experimental probability assignment

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In this lesson, you will learn how to estimate the probability of an event by using  experimental probability . Experimental probability is the ratio of the total number of times the favorable outcome occurs to the total number of trials, or times the experiment is performed. Experimental probability is helpful when you want to make predictions about something. It is important to note that the more trials you have (the more times you perform the experiment), the more accurate your prediction is likely to be. The experimental probability can be written as

Lucian used a spinner for his experiment. He created a table to display the results.

Based on the data, what is the probability that the spinner will land on 6 on the next spin?

1. Find the number of trials: 10 + 4 + 6 + 3 + 7 + 3 + 2 = 35

2. Find the number of times the event happened – The spinner landed on 6 three times.

Courtney made 15 baskets out of 25 free throw attempts. What is the experimental probability that she will make her next shot? P(basket)

1. Find the number of trials – 25 (attempts)

2. Find the number of favorable events (baskets made) – 15

3. Write the probability as a fraction, decimal, and percent. P(basket) = 3/5, 0.60, 60%

Watch the video to see more examples of experimental probability.

Experimental Probability Practice

Find each experimental probability. Write your answer as a fraction, a decimal, and a percent.

Make sure to answer with this format: 1/4; 0.25; 25%

• A high school quarterback completes 10 of 16 passes. What is the experimental probability that his next pass will be incomplete?
• Julie watches students entering the school one morning. She sees that 8 out of 25 are wearing boots. What is the experimental probability that the next student that Julie observes will be wearing boots?
• Angela makes 12 out of 16 foul shots. What is the experimental probability that she will miss her next foul shot?
• A researcher surveys 80 people to determine whether they will vote for Ford Gravitt for mayor. 42 out of 80 say ‘yes’. What is the experimental probability that the next person surveyed will say he or she plans to vote for Ford Gravitt?
• An official police radar measured the speed of 40 cars. The radar shows that 14 of the 40 cars are going faster than the posted speed limit. What is the experimental probability that the next car will be traveling at or below the speed limit?
• Nate hits an archery target 7 out of 20 times. What is the experimental probability that he will hit the target on his next try?

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• Experimental Probability – Explanation & Examples

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What is experimental probability? 

Practice questions, experimental probability – explanation & examples.

Experimental probability is the probability determined based on the results from performing the particular experiment. 

In this lesson we will go through:

• The meaning of experimental probability
• How to find experimental probability

The ratio of the number of outcomes favorable to an event to the total number of trials of the experiment.

Experimental Probability can be expressed mathematically as: 

$P(\text{E}) = \frac{\text{number of outcomes favorable to an event}}{\text{total number of trials of the experiment}}$

Let’s go back to the die tossing example. If after 12 throws you get one 6, then the experimental probability is $\frac{1}{12}$.  You can compare that to the theoretical probability. The theoretical probability of getting a 6 is $\frac{1}{6}$. This means that in 12 throws we would have expected to get 6 twice. 

Similarly, if in those 12 tosses you got a 1 five times, the experimental probability is $\frac{5}{12}$. 

How do we find experimental probability?

Now that we understand what is meant by experimental probability, let’s go through how it is found. 

To find the experimental probability of an event, divide the number of observed outcomes favorable to the event by the total number of trials of the experiment. 

Let’s go through some examples. 

Example 1:  There are 20 students in a class. Each student simultaneously flipped one coin. 12 students got a Head. From this experiment, what was the experimental probability of getting a head?

Number of coins showing Heads: 12

Total number of coins flipped: 20

$P(\text{Head}) = \frac{12}{20} = \frac{3}{5}$ 

Example 2:  The tally chart below shows the number of times a number was shown on the face of a tossed die. 

a. What was the probability of a 3 in this experiment?

b. What was the probability of a prime number?

First, sum the numbers in the frequency column to see that the experiment was performed 30 times. Then find the probabilities of the specified events. 

a. Number of times 3 showed = 7

Number of tosses = 30

$P(\text{3}) = \frac{7}{30}$ 

b. Frequency of primes = 6 + 7 + 2 = 15

Number of trials = 30 

$P(\text{prime}) = \frac{15}{30} = \frac{1}{2}$

Experimental probability can be used to predict the outcomes of experiments. This is shown in the following examples. 

Example 3: The table shows the attendance schedule of an employee for the month of May.

a. What is the probability that the employee is absent? 

b. How many times would we expect the employee to be present in June?

a. The employee was absent three times and the number of days in this experiment was 31. Therefore:

$P(\text{Absent}) = \frac{3}{31}$

b.  We expect the employee to be absent

$\frac{3}{31} × 30 = 2.9 ≈ 3$ days in June 

Example 4:  Tommy observed the color of cars owned by people in his small hometown. Of the 500 cars in town, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were grey. 

a. What is the probability that a car is red?

b. If a new car is bought by someone in town, what color do you think it would be? Explain. 

a. Number of red cars = 50 

Total number of cars = 500 

$P(\text{red car}) = \frac{50}{500} = \frac{1}{10}$ 

b. Based on the information provided, it is most likely that the new car will be black. This is because it has the highest frequency and the highest experimental probability. 

Now it is time for you to try these examples. 

The table below shows the colors of jeans in a clothing store and their respective frequencies. Use the table to answer the questions that follow.

• What is the probability of selecting a brown jeans?
• What is the probability of selecting a blue or a white jeans?

On a given day, a fast food restaurant notices that it sold 110 beef burgers, 60 chicken sandwiches, and 30 turkey sandwiches. From this observation, what is the experimental probability that a customer buys a beef burger?

Over a span of 20 seasons, a talent competition notices the following. Singers won 12 seasons, dancers won 2 seasons, comedians won 3 seasons, a poet won 1 season, and daring acts won the other 2 seasons. 

a. What is the experimental probability of a comedian winning  a season?

b. From the next 10 seasons, how many winners do you expect to be dancers?

Try this at home! Flip a coin 10 times. Record the number of tails you get. What is your P(tail)?

Number of brown jeans = 25

Total Number of jeans = 125

$P(\text{brown}) = \frac{25}{125}  = \frac{1}{5}$

Number of jeans that are blue or white = 75 + 20 = 95

$P(\text{blue or white}) = \frac{95}{125} = \frac{19}{25}$

Number of beef burgers = 110 

Number of burgers (or sandwiches) sold = 200 

$P(\text{beef burger}) = \frac{110}{200} = \frac{11}{20}$ 

a. Number of comedian winners = 3

Number of seasons = 20 

$P(\text{comedian}) = \frac{3}{20}$ 

b. First find the experimental probability that the winner is a dancer. 

Number of winners that are dancers = 2 

$P(\text{dancer}) = \frac{2}{20} = \frac{1}{10}$ 

Therefore we expect 

$\frac{1}{10} × 10 = 1$ winner to be a dancer in the next 10 seasons.

To find your P(tail) in 10 trials, complete the following with the number of tails you got. 

$P(\text{tail}) = \frac{\text{number of tails}}{10}$ 

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Theoretical and Experimental Probability Worksheets

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty.

Benefits of Theoretical and Experimental Probability Worksheets

Experimental probability is the results of an experiment, let's say for the sake of an example marbles in a bag. Experimental probability would be drawing marbles out of the bag and recording the results. Theoretical probability is calculating the probability of it happening, not actually going out and experimenting.

Download Theoretical and Experimental Probability Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

Theoretical and Experimental Probability

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Experimental Probability

Grade 7 math worksheets.

Experimental probability refers to the probability of an event based on actual experimentation or observation of outcomes.

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Experimental probability refers to the probability of an event based on actual experimentation or observation of outcomes. It is determined by conducting an experiment or observing an event multiple times and recording the number of times the event occurs.

To find the experimental probability of an event, you would divide the number of times the event occurred by the total number of trials or observations. For example, if you flipped a coin 20 times and it landed on heads 12 times, the experimental probability of flipping heads would be 12/20 or 0.6.

Experimental probability is often used in situations where it is difficult or impossible to determine the theoretical probability of an event. It can be used to estimate the theoretical probability, but it may not be as accurate as using mathematical formulas to calculate probability.

However, experimental probability can still provide valuable information about the likelihood of an event occurring, especially if the sample size is large enough to reduce the effects of randomness and variability.

Experimental Probability Examples:

Example 1: You roll a six-sided die 100 times and record the number of times each number comes up. You find that the number 3 comes up 23 times. The experimental probability of rolling a 3 on the die is therefore 23/100 or 0.23.  

Example 2: You toss a coin 50 times and record the number of times it lands on heads. You find that it lands on heads 27 times. The experimental probability of flipping heads is therefore 27/50 or 0.54.

Example 3: You draw a card from a deck of 52 cards 200 times and record the number of times you draw a heart. You find that you draw a heart 45 times. The experimental probability of drawing a heart is therefore 45/200 or 0.225.

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Experimental Probability Formula

Here are some common formulas used to calculate probability:

Theoretical Probability Formula: Theoretical probability is the probability of an event based on mathematical calculations. The formula for theoretical probability is:

P(A) = Number of favorable outcomes / Total number of outcomes

where P(A) represents the probability of event A.

Experimental Probability Formula: Experimental probability is the probability of an event based on actual experimentation or observation. The formula for experimental probability is:

P(A) = Number of times event A occurs / Total number of trials or observations

Conditional Probability Formula: Conditional probability is the probability of an event occurring given that another event has already occurred. The formula for conditional probability is:

P(A | B) = P(A and B) / P(B)

where P(A | B) represents the probability of event A given that event B has occurred, P(A and B) represents the probability of both A and B occurring, and P(B) represents the probability of event B occurring.

Multiplication Rule Formula: The multiplication rule is used to calculate the probability of two or more independent events occurring together. The formula for the multiplication rule is:

P(A and B) = P(A) * P(B)

where P(A and B) represents the probability of both A and B occurring, and P(A) and P(B) represent the probabilities of events A and B occurring, respectively.

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Experimental Probability FAQS

What is probability.

Probability is a measure of the likelihood of an event occurring. It is represented as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

What are the types of probability?

There are two main types of probability: theoretical probability and experimental probability. Theoretical probability is based on mathematical calculations, while experimental probability is based on actual experimentation or observation.

What is conditional probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated using the formula P(A | B) = P(A and B) / P(B), where P(A | B) represents the probability of event A given that event B has occurred.

What is the difference between independent and dependent events?

Independent events are events in which the occurrence of one event does not affect the probability of the other event occurring. Dependent events are events in which the occurrence of one event affects the probability of the other event occurring.

What is the law of large numbers?

The law of large numbers states that as the number of trials or observations increases, the experimental probability of an event approaches its theoretical probability. This means that with a large enough sample size, the experimental probability becomes more accurate.

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Methodology

• Random Assignment in Experiments | Introduction & Examples

Random Assignment in Experiments | Introduction & Examples

Published on March 8, 2021 by Pritha Bhandari . Revised on June 22, 2023.

In experimental research, random assignment is a way of placing participants from your sample into different treatment groups using randomization.

With simple random assignment, every member of the sample has a known or equal chance of being placed in a control group or an experimental group. Studies that use simple random assignment are also called completely randomized designs .

Random assignment is a key part of experimental design . It helps you ensure that all groups are comparable at the start of a study: any differences between them are due to random factors, not research biases like sampling bias or selection bias .

Table of contents

Why does random assignment matter, random sampling vs random assignment, how do you use random assignment, when is random assignment not used, other interesting articles, frequently asked questions about random assignment.

Random assignment is an important part of control in experimental research, because it helps strengthen the internal validity of an experiment and avoid biases.

In experiments, researchers manipulate an independent variable to assess its effect on a dependent variable, while controlling for other variables. To do so, they often use different levels of an independent variable for different groups of participants.

This is called a between-groups or independent measures design.

You use three groups of participants that are each given a different level of the independent variable:

• a control group that’s given a placebo (no dosage, to control for a placebo effect ),
• an experimental group that’s given a low dosage,
• a second experimental group that’s given a high dosage.

Random assignment to helps you make sure that the treatment groups don’t differ in systematic ways at the start of the experiment, as this can seriously affect (and even invalidate) your work.

If you don’t use random assignment, you may not be able to rule out alternative explanations for your results.

• participants recruited from cafes are placed in the control group ,
• participants recruited from local community centers are placed in the low dosage experimental group,
• participants recruited from gyms are placed in the high dosage group.

With this type of assignment, it’s hard to tell whether the participant characteristics are the same across all groups at the start of the study. Gym-users may tend to engage in more healthy behaviors than people who frequent cafes or community centers, and this would introduce a healthy user bias in your study.

Although random assignment helps even out baseline differences between groups, it doesn’t always make them completely equivalent. There may still be extraneous variables that differ between groups, and there will always be some group differences that arise from chance.

Most of the time, the random variation between groups is low, and, therefore, it’s acceptable for further analysis. This is especially true when you have a large sample. In general, you should always use random assignment in experiments when it is ethically possible and makes sense for your study topic.

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Random sampling and random assignment are both important concepts in research, but it’s important to understand the difference between them.

Random sampling (also called probability sampling or random selection) is a way of selecting members of a population to be included in your study. In contrast, random assignment is a way of sorting the sample participants into control and experimental groups.

While random sampling is used in many types of studies, random assignment is only used in between-subjects experimental designs.

Some studies use both random sampling and random assignment, while others use only one or the other.

Random sampling enhances the external validity or generalizability of your results, because it helps ensure that your sample is unbiased and representative of the whole population. This allows you to make stronger statistical inferences .

You use a simple random sample to collect data. Because you have access to the whole population (all employees), you can assign all 8000 employees a number and use a random number generator to select 300 employees. These 300 employees are your full sample.

Random assignment enhances the internal validity of the study, because it ensures that there are no systematic differences between the participants in each group. This helps you conclude that the outcomes can be attributed to the independent variable .

• a control group that receives no intervention.
• an experimental group that has a remote team-building intervention every week for a month.

You use random assignment to place participants into the control or experimental group. To do so, you take your list of participants and assign each participant a number. Again, you use a random number generator to place each participant in one of the two groups.

To use simple random assignment, you start by giving every member of the sample a unique number. Then, you can use computer programs or manual methods to randomly assign each participant to a group.

• Random number generator: Use a computer program to generate random numbers from the list for each group.
• Lottery method: Place all numbers individually in a hat or a bucket, and draw numbers at random for each group.
• Flip a coin: When you only have two groups, for each number on the list, flip a coin to decide if they’ll be in the control or the experimental group.
• Use a dice: When you have three groups, for each number on the list, roll a dice to decide which of the groups they will be in. For example, assume that rolling 1 or 2 lands them in a control group; 3 or 4 in an experimental group; and 5 or 6 in a second control or experimental group.

This type of random assignment is the most powerful method of placing participants in conditions, because each individual has an equal chance of being placed in any one of your treatment groups.

Random assignment in block designs

In more complicated experimental designs, random assignment is only used after participants are grouped into blocks based on some characteristic (e.g., test score or demographic variable). These groupings mean that you need a larger sample to achieve high statistical power .

For example, a randomized block design involves placing participants into blocks based on a shared characteristic (e.g., college students versus graduates), and then using random assignment within each block to assign participants to every treatment condition. This helps you assess whether the characteristic affects the outcomes of your treatment.

In an experimental matched design , you use blocking and then match up individual participants from each block based on specific characteristics. Within each matched pair or group, you randomly assign each participant to one of the conditions in the experiment and compare their outcomes.

Sometimes, it’s not relevant or ethical to use simple random assignment, so groups are assigned in a different way.

When comparing different groups

Sometimes, differences between participants are the main focus of a study, for example, when comparing men and women or people with and without health conditions. Participants are not randomly assigned to different groups, but instead assigned based on their characteristics.

In this type of study, the characteristic of interest (e.g., gender) is an independent variable, and the groups differ based on the different levels (e.g., men, women, etc.). All participants are tested the same way, and then their group-level outcomes are compared.

When it’s not ethically permissible

When studying unhealthy or dangerous behaviors, it’s not possible to use random assignment. For example, if you’re studying heavy drinkers and social drinkers, it’s unethical to randomly assign participants to one of the two groups and ask them to drink large amounts of alcohol for your experiment.

When you can’t assign participants to groups, you can also conduct a quasi-experimental study . In a quasi-experiment, you study the outcomes of pre-existing groups who receive treatments that you may not have any control over (e.g., heavy drinkers and social drinkers). These groups aren’t randomly assigned, but may be considered comparable when some other variables (e.g., age or socioeconomic status) are controlled for.

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If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval
• Quartiles & Quantiles
• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Prospective cohort study

Research bias

• Implicit bias
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hindsight bias
• Affect heuristic
• Social desirability bias

In experimental research, random assignment is a way of placing participants from your sample into different groups using randomization. With this method, every member of the sample has a known or equal chance of being placed in a control group or an experimental group.

Random selection, or random sampling , is a way of selecting members of a population for your study’s sample.

In contrast, random assignment is a way of sorting the sample into control and experimental groups.

Random sampling enhances the external validity or generalizability of your results, while random assignment improves the internal validity of your study.

Random assignment is used in experiments with a between-groups or independent measures design. In this research design, there’s usually a control group and one or more experimental groups. Random assignment helps ensure that the groups are comparable.

In general, you should always use random assignment in this type of experimental design when it is ethically possible and makes sense for your study topic.

To implement random assignment , assign a unique number to every member of your study’s sample .

Then, you can use a random number generator or a lottery method to randomly assign each number to a control or experimental group. You can also do so manually, by flipping a coin or rolling a dice to randomly assign participants to groups.

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Assignment: Experimental Probability

Created by Allison M (Frozen Vertex) on 09/5/2024

2 activities: 1 game, 1 assessment

Activity 1: Assessment. Estimated duration: 3 min

Assessment with 3 questions chosen from Experimental Probability.

3 questions assessment

Activity 2: Instructional Game. Estimated duration: 18 min

Probably the Greatest Artist

Our artist shows you how to throw huge numbers of color cubes in experiments. You'll learn about probability as these color cubes fall randomly. The end results are paint mixtures that you can use to create your own masterpiece of an artwork.

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Simple Probability Experiments

The topic area of probability provides a good opportunity for students to engage in practical work as a means of gaining an understanding of the basic principles of probability. This resource package contains a variety of activities providing opportunity for students to record and describe the outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes using appropriate language and use and understand the 0-1 scale.

Visit the secondary mathematics webpage to access all lists.

Handling Data 2

Quality Assured Category: Mathematics Publisher: Nuffield Foundation

This resource has nine handling data units. Unit 3 is the section of work appropriate to this topic. In  Understanding and using the probability scale  students are presented with a number of different situations in which they have to use a number line and appropriate vocabulary. In Probability using different numbers,  students use fractions, decimals and percentages to represent probabilities. Making conjectures and analysing results requires students to use the results of experiments to make predictions.

Probability

Quality Assured Category: Mathematics Publisher: SMILE

This resource contains investigations, worksheets and practical activities. The activities appropriate to this topic in probability pack one are: 

Likely or unlikely  in which students are asked to categorise probability statements. 

Experiments  where students throw a dice sixty times and record the results.

Strange dice game in which students have to decide whether or not the game is  fair.

Lucky dip in which students attempt to decribe the results from an experiment. 

Fair play in which students determine whether a game is fair. 

How likely which contains   experiments with spinners and a probability scale.

Quality Assured Category: Mathematics Publisher: Centre for Innovation in Mathematics Teaching

In  Probability students explore the use of the probability scale by considering a number of examples which are useful when introducing probability. The activity file describes a dice game in which students have to experiment to determine the probability of wining. The second activity explores a Russian fable which tells of an experiment to determine who is to be married next. The third experiment investigates the likelihood that people will have the same birthdate.

Good ideas for Probability

Quality Assured Category: Mathematics Publisher: Polygon Resources

These activities encourage students to develop their own games with appropriate time to work on them and test out their hypotheses.

Horse Race asks students to investigate whether this is a fair game.

Crossing the River involves students playing a game and then modifying it to determine whether it is now easier or harder to win and why.

21 Dice asks students to experiment with differently numbered dice to find the highest totals possible when the die is thrown and the numbers are added together.

Bingo e ncourages students to invent their own variation of the game by experimenting with other types and numbers of dice with appropriate bingo cards.

Snap involves students using a subset of cards from a full pack to play a game of snap, record their results then investigate what happens with different subsets.

Pinball Machine is an experiment to determine whether or not a particular pinball machine would produce a profit.

Roll a Penny is an experiment with grids of different sizes and the probability of achieving a desirable outcome.

Students can be placed in groups, each group investigating a diffrent game. Groups can present their findings to the rest of the class.

Probability Mystery

Quality Assured Category: Mathematics Publisher: Durham County Council

This resource has fourteen cards with statements about the likelihood of six different runners winning a race. Students are tasked with using the statements on the cards to determine who is most likely to win the race, with what probability, and in what sequence they would expect the runners to finish.

The mathematics that students need to complete the challenge is knowledge of the probability scale, to understand the probability of certainty; ‘evens chance’, likelihood and chance.

Ordering Probabilities S1

Quality Assured Category: Mathematics Publisher: Department for Education

In this resource students learn to understand that probabilities are assigned values between 0 and 1. Students decide an appropriate value for the probability of a given event and use some of the vocabulary associated with probability such as ‘certain’, ‘impossible’, ‘likely’. They will develop their ability to order decimals between 0 and 1. Students should have some experience of decimal numbers.

Evaluating Probability Statements S2

In this resource students discuss and clarify some common misconceptions about probability. This involves discussing the concepts of equally likely events, randomness and sample sizes. Students also learn to reason and explain. This session assumes that learners have encountered probability before. It aims to draw on their prior knowledge and develop it through discussion; it does not assume that they are already competent.

Quality Assured Category: Mathematics Publisher: The Virtual Textbook

This resource contains four interactive excel files dealing with:

Probability Line and Simple Examples This excel file begins with the probability number line labelled with fractions, decimals and percentages. The next two interactive sheets pose problems about selecting a coloured ball from a bag with answers shown as a fraction, simplified fraction and percentage. Other sheets pose questions about selecting a letters from a given word and rolling a single die. The final interactive sheet shows the possibility space when two dice are thrown and the answers to the questions posed are shown as fractions and simplified fractions.

There are a further five sheets of questions which may be suitable for use in the classroom.

Probability:  Biased  Coins This short interactive program illustrates the effect of bias when a number of coins are tossed. The first sheet deals with the experiment when 10 coins are tossed when the probability of getting a head can be altered. The probability of getting a particular number of heads is shown in a table (alongside the cumulative probability) and in a bar chart. The second interactive sheet deals with a similar experiment when 20 coins are tossed.

There are three more sheets of questions which may be suitable for use in the classroom.

Probability: Relative Frequency This excel file simulates selecting an object from a given number of objects in a bag. The outcomes are shown in a tally chart and the problem is to predict how many of each coloured object there are in the bag. The contents of the bag can be revealed on a separate sheet.

Probability: Two Events This interactive file starts gently with a sheet showing the 52 cards in a standard deck. Questions are posed about the probability of selecting a single card and answers can be revealed. The next two sheets deal with probabilities associated with selecting two cards, one with replacement and the other without replacement. The outcomes are shown on tree diagrams and the probabilities on each branch can be revealed. The problem of rolling two dice is similarly dealt with on the next two sheets with one of them using biased dice. The final interactive sheet uses a tree diagram to illustrate the solution to problems related to selection of objects from two different bags.

There are a further eight sheets of questions which may be suitable for use in the classroom.