Aim: Permution/Combination Problems Course: Math Lit. Do Now: Aim: How do we use permutations and...

Aim: Permution/Combination Problems Course: Math Lit.

Do Now:

Aim: How do we use permutations and combinations to solve 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.


With permutations

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

5P5

6P6

= 1/61P1 •

6P6Number of ways 6 seats could be occupied

5P5Number of ways other 5 seats are occupied

1P1Number of ways for Ed to side in

1st seat

Permutations & Probability problems

P En En S

( )( )( )

1

( )6

P E


Model Problem

Dependent Events

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?

P(first 5) = 4/52 P(second 5) = 3/51P(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

= 1/221

4C2 52C2


Probability Involving Combinations

Dependent Events

2 Permutations

1 Counting Principle

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?

P(1st 5) = 4/52 P(2d 5) = 3/51P(5, 5) = 4/52 • 3/51

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

4P2

52P2

= 1/221

3 Combinations 4C2

52C2

= 1/221


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?

P En En S

( )( )( )

P En En S

( )( )( )

!r

PC rn

rn !r

PC rn

rn

P(2 s, 2 j) =n(4-member

combinations made from the 9 members)

n(combination of 2s & 2j)

Permutation or

Combination?


Model Problem

Successful Outcomes n(E):

Total Outcomes n(S):Nine total students from

which to choose


: 9C4 = 126

How do we determine n(2 soph., 2 jun.)?

Two of four sophomores 4C2 = 6Two of five juniors 5C2 = 10

n(2 soph., 2 jun.) = 4C2 • 5C2 = 6•10 = 60


Model Problem

P(2 sophs., 2 juniors) =


9C4

4C2 • 5C2

= 6 • 10 126

= 60 126

= 1021


Model Problem

Total Outcomes n(S) :

P(at least 2 b) =

5C2

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?

9C3

At least 2 are blue: (2-b, 1-w) or (3-b)

Successful Outcomes n(E) 5C2 • 4C1 + 5C3

4C1• 5C3

9C3Total Outcomes n(S)

=

84

40 + 10= 25/42

Successful Outcomes n(E)

P(A B)

= P(A) + P(B) - P(A B)


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(2R) = 3C2 P(2G) = 7C2

10C2 10C2

P(A B) = P(A) + P(B) - P(A B)

P(2R or 2G) = = 8/15= 3 + 214510C2

3C2 + 7C2


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?

Aim: Permution/Combination Problems Course: Math Lit. Do Now: Aim: How do we use permutations and...

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