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?

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:

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

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.

### 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

- Discuss favourite pizzas.
- Pose the problem.
- Brainstorm ways to approach the problem that include finding all possible pizzas.
- 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? - 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 of all the possible two-topping combinations. Such a list might be organised like:

Ham and pineapple

Ham and tomatoes

Ham and peppers

Ham and mushrooms

Ham and onions

Pinepple 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.

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