Penny's Pizza

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Purpose

This problem solving activity has a statistics focus.

Achievement Objectives
S4-4: Use simple fractions and percentages to describe probabilities.
Student Activity

Decorative image of a pizza.Penny's favourite pizza restaurant offers 6 toppings: ham, onions, mushrooms, pineapple, tomato and peppers.
Penny orders a pizza with ham and pineapple.
Unfortunately the server only writes down that she wants 2 toppings but doesn't write down what they are.
The cook decides to pick two toppings at random.

What is the probability that Penny will get the pizza she ordered?

Specific Learning Outcomes
  • Make a list of possible outcomes as a method of finding probability.
  • Express the outcome as a fraction.
  • Devise and use problem–solving strategies to explore situations mathematically (make an organised list).
Description of Mathematics

This problem involves working out the probability of an event when the outcomes are equally likely to occur. (In this case each topping is equally likely to be chosen by the chef.) The probability of any particular outcome is:
1 divided by the total number of outcomes.
 

In the problem extension we are interested in a particular set of outcomes. The probability is expressed by the fraction:
A set of outcomes divided by the total number of outcomes.

Students should see that any fraction of this type must be between 0 and 1, since the numerator cannot be negative and cannot be bigger than the denominator. If the set of "outcomes that you are interested in" contains all the possible outcomes, then you are certain to get the result you are interested in, and the fraction is equal to 1. Likewise, an event that is impossible has a probability equal to 0.

The focus is on making a list of possible outcomes as a method for finding probability.

Activity

The Problem

Penny's favourite pizza restaurant offers 6 toppings: ham, onions, mushrooms, pineapple, tomato and peppers. Penny orders a pizza with ham and pineapple. Unfortunately the server only writes down that she wants 2 toppings but doesn't write down what they are. The cook decides to pick two toppings at random.

What is the probability that Penny will get the pizza she ordered?

Teaching Sequence

  1. Discuss favourite pizzas.
  2. Pose the problem.
  3. Brainstorm ways to approach the problem that include finding all possible pizzas.
  4. As students work with a partner on the problem, ask:
    How do you know that you have found all the pizzas?
    How have you organised your search for the pizzas?
    How many of these pizzas are what Penny ordered?
  5. Share solutions.

Extension

Penny actually likes all the toppings except peppers and mushrooms. What is the probability that she will get a pizza that she likes now?

Solution

Probably the most common approach is to make a list or tree diagram of all the possible two-topping combinations (ensuring that all combinations are only recorded once). Such a list might be organised like:

Ham and pineapple
Ham and tomatoes
Ham and peppers
Ham and mushrooms
Ham and onions

Pineapple and tomatoes
Pineapple and peppers
Pineapple and mushrooms
Pineapple and onions

Tomatoes and peppers
Tomatoes and mushrooms
Tomatoes and onions

Peppers and mushrooms
Peppers and onions

Mushrooms and onions

With the list of 15 pizzas (5 + 4 + 3+ 2 + 1) , the students will be able to see that Penny has 1/15 chance of getting the pizza she ordered.

Solution to the Extension

6 of the 15 pizzas are ones that Penny likes, so her probability of getting one of these is 6/15.

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Level Four