Penny's Pizza
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)
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 ordered a pizza with ham and pineapple. Unfortunatelythe server only wrote down that she wanted 2 toppings but didn’t write down what they were. The cook decided to pick two toppings at random.
What is the probability that Penny will get the pizza she ordered?
Teaching sequence
- Discuss favourite pizza’s as a way to gain interest in the problem.
- Read the problem with the class.
- Brainstorm ways to approach the problem. At this point you might be encouraging the students to plan ways to find all possible pizzas.
- Students work with a partner on the problem.
How do you know that you have found all the pizzas?
How have you oraganised 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 pizza’s (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 Penny there are 6 of the 15 pizza’s that Penny likes so her probability of getting one of these is 6/15.
| Attachment | Size |
|---|---|
| PennyPizza.pdf | 77.81 KB |
| PennyPizzaMaori.pdf | 83.13 KB |
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