1. Aim: How do we use permutations and combinations to solve probability problems?
Do Now:
Six students are arranged at random on a bench. What is the probability that Ed, one of the students, is in the first seat if the bench sits six.

2. Permutations & Probability problems
Six students are arranged at random on a bench. What is the probability that Ed, one of the students, is in the first seat if the bench sits six.
W/O permutations
P(Ed in first seat) = 1/6
With permutations
5P5
Number of ways other 5 seats are occupied
1P1
Number of ways for Ed to side in 1st seat
5P5
6P6
= 1/6
1P1 •
6P6
Number of ways 6 seats could be occupied

3. Model Problem
Two cards are drawn at random from a standard deck of 52 cards, without replacement. What is the probability that both cards drawn are fives?
Dependent Events
P(first 5) = 4/52
P(second 5) = 3/51
P(5, 5) = 4/52 • 3/51
= 1/221
Number of ways to draw 2 5’s from possible 4
Number of ways to draw any 2 cards from deck of 52
4C2
52C2
4C2
52C2
= 1/221

4. Probability Involving Combinations
Two cards are drawn at random from a standard deck of 52 cards, without replacement. What is the probability that both cards drawn are fives?
Dependent Events 1
Counting Principle
P(1st 5) = 4/52
P(2d 5) = 3/51
P(5, 5) = 4/52 • 3/51 2
Permutations
4P2
52P2
= 1/221 3
Combinations
4C2
52C2
= 1/221
Number of ways to draw 2 5’s from possible 4
Number of ways to draw any 2 cards from deck of 52
4C2
52C2

5. Permutation or Combination?
Model Problem
In a school organization, there are 4 sophomores and 5 juniors. A committee of 4 people is to be selected from this group. What is the probability that 2 sophomores and 2 juniors will be on the committee?
Permutation or Combination?
n(combination of 2s & 2j)
P(2 s, 2 j) =
n(4-member combinations made from the 9 members)

6. Nine total students from which to choose
Model Problem
In a school organization, there are 4 sophomores and 5 juniors. A committee of 4 people is to be selected from this group. What is the probability that 2 sophomores and 2 juniors will be on the committee?
Total Outcomes n(S):
Nine total students from which to choose
: 9C4
= 126
Successful Outcomes n(E):
How do we determine n(2 soph., 2 jun.)?
Two of four sophomores
4C2
= 6
Two of five juniors
5C2
= 10
n(2 soph., 2 jun.) = 4C2 • 5C2 = 6•10 = 60

7. Model Problem
In a school organization, there are 4 sophomores and 5 juniors. A committee of 4 people is to be selected from this group. What is the probability that 2 sophomores and 2 juniors will be on the committee?
4C2 • 5C2
P(2 sophs., 2 juniors) =
9C4
= 6 • 10
126
= 60
126
= 10 21

8. Successful Outcomes n(E)
Model Problem
An urn contains 4 white marbles and 5 blue marbles, all of equal size. Three marbles are drawn at random with no replacement. What is the probability that at least 2 marbles drawn are blue?
Total Outcomes n(S) :
9C3
Successful Outcomes n(E)
At least 2 are blue: (2-b, 1-w) or (3-b)
P(A  B)
= P(A) + P(B) - P(A  B)
5C2 •
4C1
5C3
Successful Outcomes n(E)
5C2 • 4C1 = +
5C3
9C3
Total Outcomes n(S)
P(at least 2 b) = 84
= 25/42

9. Model Problem
A candy dish contains 10 candies. Three candies are covered with red foil and 7 with green foil.
A. If 2 candies are chosen at random from the dish, what is the probability that both will be covered with the same colored foil?
P(both red or both green) = P(2R) + P(2G)
P(A  B) = P(A) + P(B) - P(A  B)
P(2R) = 3C2
P(2G) = 7C2
10C2
10C2
P(2R or 2G) =
10C2
3C2 + 7C2 = 45
= 8/15

10. Model Problem
A candy dish contains 10 candies. Three candies are covered with red foil and 7 with green foil.
B. If 2 candies are chosen at random from the dish, what is the probability that each will be covered with a different foil?