This document provides examples of using permutations and combinations to calculate probabilities of events occurring. It discusses: - Calculating the probability of selecting two white and two red balls from an urn containing six white and five red balls by finding the number of ways to choose two white and two red balls, and dividing by the total number of possible outcomes. - Calculating probabilities for other events involving selecting balls from urns with different ball combinations. - Calculating probabilities for events involving selecting students for a team from groups with different numbers of boys and girls. - Calculating the probability of choosing a specific letter from a word by counting the number of ways to choose that letter and dividing by the total letters.

1. Probability Using Permutations or Combinations
Example: An urn contains six white balls and five red balls.
A sample of four balls is selected at random from the urn.
What is the probability that the sample contains two white
balls and two red balls.
From our definition of probability we know:
P(E) = # of ways event can occur
Total number of possible outcomes
Step #1: Find the number of ways event can occur or
another way of saying it.......
How many ways can we choose two white balls and two red
balls? 6C2 *5C2 = 150
Step #2: Find the total number of possible outcomes or
another way of saying it.........
How many ways can we select four balls out of 11 balls?
11C4 = 330
Therefore, the probability is: 150 or 5
330 11

2. Example:
An urn contains six red balls and four green balls.
A sample of seven balls is selected at random. Find
the probability that:
a. five red and two green balls are selected
b. all the balls selected are red
c. more than four red balls are selected.

3. Example:
There are seven boys and three girls on a school tennis
team. The coach must select four people from this group to
participate in the county championship?
a. How many four­person teams can be formed from the
group of ten students?
b . In how many ways can two boys and two girls be chosen
to participate in the county championship?
c. What is the probability that two boys and two girls are
chosen for the team?

4. Example: If one of the letters of the word
'ELEMENT' is chosen at random, what is the
probability that an E is chosen?
Example: A bag contains three chocolate, four sugar, and five
lemon cookies. Greg takes two cookies from the bag for a snack.
Find the probability that Greg did not take two chocolate cookies
from the bag. Explain why using the complement of the event of
not choosing two chocolate cookies might be an easier approach to
solving this problem.