Counting on probability

1.

We begin the unit by playing the game Odds and Evens. This uses a spinner divided into thirds with the segments labelled 1, 2, or 3. You can use Counting on probability odds and evens or make your own.

Odds and evens

Players (in two teams or pairs) take turns spinning the spinner twice.

One player (or team) wins if the sum of the two numbers is even.

The other player or team wins if the sum is odd.

2.

Before the students start playing the game, ask:

• What does the word fair mean? (some notion of balance in the chance of winning)
• Do you think this is a fair game?
• Why or why not?
• How might we work out if the game is fair?

3.

Look for students to express ideas about each player having the same chance of winning. Some students might suggest finding out all the things (outcomes) that might happen and assigning a win or loss to the appropriate player.

4.

Let the teams or pairs play the game, keeping track of the number of wins for Player Odd and for Player Even.

Ten games per pair of students will generate a lot of experimental data (i.e., data that comes from the results of an experiment). Discuss the results and the students’ ideas about fairness. Record some of the results in a table:

5.

Discuss the experimental results.

• What do you notice about the results? (Students should notice that the results vary a lot as the sample of ten games is quite small.)
• Why does this variation happen? Is it expected? (Students might offer ideas about the consistency of the spin, the unpredictability of chance, and the small number of games.)
• How did you decide if the game was fair?
• Why do some think it is fair and others think it isn’t fair? Can it be both?
• Are we better to leave the data in team groups like this or combine it? What difference might combining the data make?

6.

You might pool the class data and get a result like: Odd (69 wins), Even (81 wins).

• Is the game fair? How can we tell? (There may be a minor imbalance in favour of even.)

7.

Deciding if the game is fair or not should lead to the idea that to predict fairness, you need to know all possible outcomes, that is, all possible totals of numbers on the two spinners. From this, you can see how many are odd and how many are even.

• How could we find all the possible outcomes?

Share ideas.

8.

Model the creation of a tree diagram for all the possible outcomes. You might start the diagram and expect students to complete it, or model the creation of one tree diagram and then ask students, in pairs, to create one.

• First, list the possible outcomes of the first spin (1, 2, 3).
• Next, decide how many outcomes are possible for the second spin and draw that number of branches from each of the outcomes of the first spin.
• Next, calculate the sum at the end of each branch.
• The sums can now be separated into even and odd numbers, giving 5 even and 4 odd. Therefore, the game is not fair. Over a long series of trials (games), you would expect to get an even sum more often than an odd one.
• Record the probabilities as fractions: probability of Even total = 5/9, and probability of Odd total = 4/9.

9.

Discuss:

• What fraction does 5/9 + 4/9 add to? (9/9, which equals 1).
• Why 1? (The whole in this situation is the total number of outcomes, 9, and those outcomes are split between Even win and Odd win.)

10.

Conclude by asking how the game might be altered to make it fair. Students might suggest a spinner with numbers 1, 2, 3, and 4 in quarters.

• Would that work? Show all the possible outcomes of two spins on that spinner.
• Students might also suggest changing some rules.
• Would taking three spins instead of two make any difference to fairness?

Over the next 2–3 days, give the students the following problems to solve.

Adapt the problems to relate to the cultures, interests, and context reflected in your class. You might introduce the problems lesson-by-lesson or set them out as stations to visit. In either case, ensure that a plenary is held at the end of each lesson to share ideas. You should provide modelling, feedback, and small-group and/or individualised teaching to support students in successfully engaging in all of the problems presented. Encourage students to justify their solutions and their choice of method for counting all the possible outcomes. Ask students to also record the probabilities associated with each of the problems as fractions (decimals and percentages as extensions).

Printable versions of the problems are available as Counting on probability word problems. You can also download Counting on probability word problem answers.

Problem 1: Sam’s sandwich

Every morning, Sam makes his own sandwich. He can use brown or white bread and can choose between honey, cheese, Vegemite, lettuce, or tomato.

• How many different sandwiches can Sam make with one filling?
• If Sam makes each possible sandwich equally often, what is the probability that he makes a white sandwich with honey for lunch today?

Problem 2: Pete’s pizza

At Pete’s Pizza, there is a special on. You can choose two toppings for your pizza. One single choice has to be made from the ham, mushroom, pepperoni, and bacon bins, while the other single choice has to come from the pineapple, tomato, and extra cheese bins.

• How many different pizzas are possible with two toppings?
• If people choose the toppings at random, what’s the probability that Pete will sell a bacon and tomato pizza next? (Random means that the choice of each topping is equally likely.)

Problem 3: Coin tossing

Georgia is the captain of her netball team and always selects heads at the start of the game.

• What is the probability that she will win the toss for the next four games?

Problem 4: Licence plates

In Botutuland, car licence plates have two letters and four digits.

• If all of the 10 digits can occur in any one of the four spots, how many different licence plates are possible?
• If all the licence plates are being used, what’s the probability of a Botutulandian seeing a licence plate with 0000?

Problem 5: Travel

Laura can travel from her home to the city by train, ferry, or bus. Then she can walk, taxi, or cycle from the transport terminal to her office.

• How many different ways can she travel from her home to her office?
• If she is just as likely to select any of the transport types, what is the probability that she will travel by ferry and then walk to her office?

Problem 6: Sports

At Tim’s school, students can choose one sport from cricket, surfing, tennis, or touch rugby in the summer. They can choose one sport from soccer, hockey, rugby, or basketball in the winter.

• How many different sports combinations are possible?

Imagine that players drew the sports randomly, like out of two hats, and each sport had the same chance of selection.

• What is the probability that Tim plays only sports that involve a round ball?
• What is the probability of Tim not playing rugby?

Problem 7: Spinners II

Mala has a spinner that is divided into quarters, with the numbers 1, 2, 3, and 4 separately on each quarter. She plays the odd and even game with her friend Erik.

• Is this a fair game?
• What event(s) has a probability of 10/16? (An event is a particular outcome set, like an odd total, an even total, or a total greater than three.)

Problem 8: Spinners III

Mala has a spinner that is divided into quarters, with 1s in two of the opposite quarters and 2 and 3 in the other two quarters. She plays the odd and even game with her friend Erik.

• Is this a fair game?

Problem 9: Spinners IV

Mala has a spinner that is divided into quarters, with 1s in two of the opposite quarters and 2 and 3 in the other two quarters. She and her friend Erik spin the spinner twice and add the scores of the two spins. In this game, you win if you get a total that is divisible by 2.

• Is this a fair game?

Problem 10: Spinners V

Mala has a spinner that is divided into quarters, with 1s in two of the opposite quarters and 2 and 3 in the other two quarters. She plays the following game with her friends Erik and Nelio. The scores of two spins are added. Mala wins if the sum is divisible by 3; Erik wins if the sum has a remainder of 1 when divided by 3; and Nelio wins if the remainder is 2 when the sum is divided by 3.

• Is this a fair game?

Problem 1 Answer

Sam has the choice of 2 breads and 5 fillings. He has the choice of 2 x 5 = 10 different sandwiches. This can be solved using a tree diagram that first has 2 branches (one for each of the bread types) and then 5 branches at the end of the first branches (one for each of the fillings). This will give 10 ends to the tree.

Now, a white sandwich with honey is just one of the ten possible sandwiches. So the chances of Sam making that particular sandwich is 1/10.

Problem 2 Answer

For the pizza, you get a choice of 4 from one bin and 3 from the other. Hence, there are 12 possible pizzas with two toppings, one topping coming from each bin.

The bacon and tomato combination is just one of the 12 possible pizzas, so the chances of that pizza are 1/12.

Problem 3 Answer

If you construct the tree here, you start off with two branches, and you add two branches at the end of each branch until four lots of branch levels have been added. This gives 16 possible outcomes. Only one of these is all heads. So the probability that she wins four calls in a row is 1/16.

Problem 4 Answer

It should be pointed out that this problem is a little hard to solve using a tree diagram. First of all, there are 10,000 possible numbers that can be used (going from 0 to 9999). Now there are 26 letters that can go in the first letter position and 26 for the second.

Altogether, there are 26 x 26 x 10,000 = 6,760,000 number plates.

Now there are 26 x 26 = 676 number plates with all zeros. So the probability of seeing one of these is 676/6,760,000 = 1/10,000. (You probably wouldn’t see one of these very often in Botutuland.)

Problem 5 Answer

Laura has 3 ways of getting to the city and then 3 ways of getting to the office. Therefore, she has 3 x 3 = 9 ways of getting to work.

The probability of going by ferry and then walking is 1/9.

Problem 6 Answer

Tim has 4 x 4 = 16 choices of sports. He has a choice of 2 round-ball games in summer and 3 in the winter, for a total of 6 combinations. Hence, his chances of playing a round-ball game is 6/16 = 3/8

There are 3 sports that are not rugby in the summer and 3 sports that are not rugby in the winter. Since 3 x 3 = 9, he has a 9/16 chance of not playing any rugby.

Problem 7 Answer

There are 16 outcomes here, and 8 of them are even. The game is fair.

One way of getting a probability of 10/16 is to take the event "the sum is less than 6". But "more than 3" works equally well. But there are other possibilities.

Problem 8 Answer

Here, 10 of the 16 outcomes are even. So it is best to be even. This is definitely not a fair game.

Problem 9 Answer

There are only 5 outcomes here out of 16 that have a number divisible by 3. Hence, this is not a fair game.

Problem 10 Answer

This is not a fair game, either. There are 5 sums divisible by 3; 5 sums with a remainder of 1 when divided by 3; and 6 with a remainder of 2 when divided by 3. So it is better to be Nelio.

Incidentally, since there are 16 outcomes here and 16 is not divisible by 3, there is no way that this game could be fair.

In today’s session, the students work in pairs to make up their own outcomes, counting problems for other students to solve.

1.

Introduce the session by discussing the variety of T-shirts worn (or other clothing items) by class members. List the different ways that T-shirts can vary, for example, in colour, size, logo (or not), and neck style. These features are variables in the design of a t-shirt because they can change.

2.

Ask the students to work in pairs to make up a T-shirt problem using 3 of the variables listed. For example, Aroha designs t-shirts in four colours: white, black, red, and green. She produces three sizes: S, M, and L. Her t-shirts can have either a V-neck or a crew neck.

• How many different designs can Aroha make? (4 x 3 x 2 = 24)
• If you reach into Aroha’s blindfold bin, what are the chances that you will get a large, white t-shirt? (2/24 or 1/12)

3.

Tell them to write the problem on one side of a piece of paper (or on the outside cover of a folded piece) and the solution on the other side.

4.

Then get them to trade problems with other students. When they have solved the problem, compare their solution with that of the problem writers.

5.

Leave the problems on display on the maths board or make a book of outcome problems.

6.

Students might enjoy exploring the creation and trialing of other spinners. Go to the Spinners learning object to try this out.