Learning Resources

Lesson

We now focus again on calculating probability. Recall that for an event X, the probability of X occurring is given by the formula:

In the following examples the "total number of possible outcomes" and sometimes the "number of ways of X  occurring" involve calculating combinations and/or permutations.

You should be familiar with the concepts of probability, and you should also be comfortable working with combinations and permutations. We will now look at using permutations and combinations to calculate probability. This is best shown by working through some examples.

Example 1 

A regular deck of 52 playing cards is well shuffled. You are dealt 5 cards. What is the probability you receive three aces and two eights?

In the above example, both the number of ways that event X could occur and the total number of outcomes were combinations. It some examples one or both of these could be permutations. Consider the following example.

Example 2

The security system on a building has a 4 digit code that must be entered to deactivate the alarm when the door is first opened in the morning. The pad on which the code is to be entered contains the10 digits 0 through 9.  What is the probability that a thief could randomly enter the correct code and deactivate the alarm if (1) no digit is allowed to be used more than once in the code, (2) digits are allowed to be repeated in the code.

Solution

(1) There is only one correct code, hence the number of outcomes which favour
      event X is 1. The total of all possible outcomes is a permutation of the 4
      digits chosen from a set of 10, i.e. ₁₀P₄, which from the formula is:

 

Thus the probability of randomly deactivating the alarm is 

(2) Again there is only one correct code so the number of outcomes which favour event X is 1. The total of all possible outcomes can be found by using the multiplication principle. Since repetition is allowed, there are 10 different choices for each digit. Thus the total number of all possible outcomes is:

10 x 10 x 10 x 10 = 10000

Thus the probability of randomly deactivating the alarm is 

Activity

1. Study Examples 8 and 9 in Focus H on pages 336 & 337 in your text.
2. Complete the Focus Question 1 on page 337.
3. Do the CYU Questions 2 - 9 (b) on pages 337 - 339.

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

A local group is raising funds to send the school team to a national tournament. They devise a lottery, similar to 6/49, that requires you to choose 4 numbers from 1 to 20. If you buy one ticket, what is the probability of winning this lottery?

Solution