Probability Distributions

Purpose

This unit aims to introduce students to concepts that are important in probability. The emphasis should be on working slowly taking the time to let the students grasp the ideas that underlie the activities presented. All the ideas are important as a base for future learning in this area of mathematics.

Specific Learning Outcomes
• explain what a probability distribution is
• explain how probabilities are distributed
• calculate what happens to the distribution when variables are summed
• calculate expected number from theoretical probabilities
• appreciate the difference between theoretical and experimental probabilities.
Description of Mathematics

The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values. It is also sometimes called the probability function or the probability mass function. If we were to toss two coins, we know that there is an even chance of getting heads or tails for each. In this case, the probability distribution of the event "tossing two fair coins" looks like this: Two tails, One head and one tail, Two heads 1/4 1/2 1/4. There is a one in four chance that both coins will be tails, a one in 2 chance that one coin will be tails and the other tails, and a one in four chance that both coins will be tails.

Required Resource Materials

Dice

Coins

Spinners

Key Vocabulary

probability distribution, random variables, uniform variables, simulation

Activity

Session 1

This session aims to give the students a feel for what a probability distribution is, and to investigate the sum of 2 variables.

1. Group the class in 3s.
2. Issue each group with a pair of dice and pose a contextual question that will form the scaffolding for the skills and ideas they will cover in this session. If you are playing a board game that involves rolling two dice and moving the sum shown, what is the probability that your next move would be a move of 8 spaces?
3. They investigate the sum of the two numbers when the pair of dice is rolled (difference is simply larger minus smaller).
What all the possible sums from the two dice?
Are any sums more likely than others? How do you know?
4. Ask the students to use list the sample space to find the probability distribution. They may need to be shown how to set their working out in a 6x6 table, or they may have their own orderly method of listing the sample space.
5. Pose the following questions to groups. What is the probability that rolling two dice gives a sum of 8? Which sum(s) from the two dice has the highest probability? Which sum(s) from the two dice has the lowest probability?
6. Make a spinner that simulates, in one spin, the sum of rolling two dice.
7. Consolidate students' learning with some or all of the following activities, as time allows. For each of the following pairs of variables (1) find the distribution of the sum (2) show the distribution as both a table and a graph (3) draw a spinner that simulates the sum in a single spin. Session 2

This session uses the knowledge and technique developed in session 1 to solve word problems.

1. Pose the following problems for the class to solve in small groups. Ask the students for their ideas on how to proceed.
Problem 1: Three seeds are planted. Each has probability 0.3 of growing. Draw the distribution of the number of seeds that grow.
Problem 2: Three lightbulbs are switched on for a month. The chance of any one of the bulbs blowing during this period is 0.2. Find the probability that all will blow during the period; 2 will blow; 1 will blow; none will blow. Graph the distribution.
Problem 3: It is known that 10% of torches made by a particular factory are defective. If you buy 3 torches, what is the probability distribution for the number of them that will be faulty?
2. Share methods and solutions.
3. If time, get the students to make up their own problem(s) along these lines, for each other to solve.

Session 3

This session has aims: to investigate the distributions when uniform variables are added, and to add 3 variables.

1. In groups add the following pairs of uniform variables. Discuss the images below. Have the students suggest possible scenarios that would lead to such a distribution. What are the possible values of each variable? Give me an example of a situation that would have an even chance of scoring a 1 or a 2 or a 3. What are the possible values of the sum of the two variables? Give me an example of what might be happening here? Try to encourage the class to come up with a wide variety of possible contexts, no matter how wacky, placing an emphasis on their scenario having numerical accuracy. 2. Graph the distribution of the sum of the two variables in each of the above images.
3. Give the student a context based problem to think about for now and to work with and solve in the last session. I've invented a new game that involves three dice. I've marked each die with only a 1, 2 or 3 on the faces in such a way: The first has sides 1,1,1,2,2,3 and the second has 1,1,2,2,3,3 and the last has 1,2,2,3,3,3. Each player rolls the three dice and records their result. What is the most likely sum of the three variables?
4. Class discussion. How would you add the 3 variables in the diagram below? Find the probability distribution.
Introduce the probability tree diagram as a method of finding the probability distribution. 5. Pose the following questions for a class discussion: How would you add the 3 variables in the example below? What is the probability that the sum of the three numbers is 4? Share strategies and approaches: 6. Ask the students to, in pairs, complete the previous example and draw the distribution of the sums.

Session 4

This session aims to give the students a feel for using a probability distribution, to set up an run a simulation, to solve a problem involving expected number.

1. Group the class in 3s.
2. Issue each group with a pair of dice and pose a contextual question that will form the scaffolding for the skills and ideas they will cover in this session. If four friends are playing a two dice board game that requires each player to roll a double (difference of zero) before they can start, what is the expected number of rolls for all four of the players before everyone has started?
3. They can solve the context problem by investigating the difference between the two numbers when the pair is rolled (difference is simply larger minus smaller).
What does investigation mean in this situation?
What all the possible answers?
Are some more likely than others? How do you know?
Which are most likely?
Which are least likely?
What is the probability of each outcome (that is, what is the probability that the difference will be 0, 1, 2, 3, 4, 5)?

Leave students to their own methods.
4. Discuss the results as a class:
What answers did you find?
How did you go about it?
5. If no one suggested a 6x6 table show them how one can be constructed.
 1 2 3 4 5 6 1 0 1 2 3 4 5 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 5 4 3 2 1 0 1 6 5 4 3 2 1 0
6. From the table the probability distribution can be found.
 0 1 2 3 4 5 1/6 5/18 2/9 1/6 1/9 1/18 7. Discuss the context question (2) with the class, noting the difference between probability and expected number.
8. Set up a simulation to model the context question (2). For this, random numbers from 1-36 could be used (on a calculator, 36RAN# +1) and each student in the class should run the simulation until every player is on the board. (Note that once a player is on the board, they will continue to roll the dice when it is their turn).
9. Regroup and review the simulation. Find the mean of the students' results for the number of rolls it took before all players were on the board (including players on the board taking their turn). Use this result to compare theoretical and experimental probabilities and to compare probabilities with expected number.

Session 5

This session uses the knowledge and technique developed in the previous to solve word problems independently.

1. Pose the following problems for the class to solve in small groups. Ask the students for their ideas on how to proceed. Encourage the students to calculate a theoretical value, then to set up and run a simulation to generate an experimental value.
Problem 1: A gardener is making a 'wildflower bed' by scattering 100 lupin seeds, 100 cosmos seeds and 50 fox gloves seeds. He then scatters 1/5 of the seeds in the flower bed. If half of the seeds planted are successful, what is the expected number of fox gloves in the bed?
Problem 2: A factory produces a toy that has three parts. The probability of any one part having a fault is 1%. If the factory produces 1500 toys per day, what is the expected number of faulty toys (one or more faulty part) produced in a day?
2. Problem 3: (Recall the probability problem discussed in session 3). I've invented a new game that involves three dice. I've marked each die with only a 1, 2 or 3 on the faces in such a way: The first has sides 1,1,1,2,2,3 and the second has 1,1,2,2,3,3 and the last has 1,2,2,3,3,3. Each player rolls the three dice and records their result. (And now for how to win!) Each player takes a turn and adds their new score to their running total. The winner is the first to break 66 (ie their running total goes over, not including 66). Find the expected number of throws for a player to reach 66.
3. Bring the class together to combing experimental values (and find the class mean of those values). Compare this mean with the theoretical value. Discuss.
4. If time, get the students to make up their own problem(s) along these lines, for each other to solve.

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