11.5 Probability with the Fundamental Counting Principles, Permutations and Combinations

Three Kinds of Probability

• Theoretical Probability• Empirical Probability• Subjective Probability

Probability with FCP (Problem # 1)

• A restaurant offers 3 appetizers (salad, soup, breadsticks), 4 main courses (chicken, beef, fish, pork) and 2 desserts (cake and ice cream). What is the probability of getting a soup, chicken and ice cream for a meal?

1/24

Probability with FCP (Problem # 2)

• Mr. P has 4 ties (Cartoon character, Navy Blue, Black, Striped), 4 shirts (white, blue, black and gray) and 4 pairs of pants (black, brown, beige and gray). What is the probability that he will wear a black tie, black shirt and black pants?

• 1/64

Probability of a Permutation

Number of ways the permutation can occur Total number of permutations

Probability with Permutations (Problem # 3)

• Jay-Z, Jeremih, R. Kelly and T.I. were invited to the 50th celebration at Seton. They arrived randomly and each person arrived at different time.

• In how many ways can they arrive?

• In how many ways can Jay-Z arrive first and T.I. last?

• What is the probability that Jay-Z arrive first and T.I. last?

Answer:

a. Total Permutation: 4! = 24

b. No. of permutations for Jay Z to arrive first and T.I. last = 1 x 2 x 1 x 1 = 2

c. Probability = 2/24 = 1/12

Probability with Permutations (Problem # 4)

• Suppose that you want to arrange the 4 Twilight books.

a. What is the probability that Breaking Dawn (the last book in the series) is placed at the end?

b. What is the probability that Twilight (the first book in the series) is placed first and New moon (the second in the series) placed second?

A. 6/24 or ¼B. 2/24 or 1/12

Probability with Permutations (Problem # 5)

• Miley Cyrus, Taylor Swift, Demi Lovato, Lady GaGa and Kelly Clarkson agree to hold a concert. What is the probability that Lady GaGa will perform first and Kelly Clarkson last?

• 6/120 = 1/20

Probability of a combination

Number of ways the combinations can occur Total number of combinations

Probability with Combinations (Problem # 6)

• A poker hand consists of five cards.

• a. Find the total number of 5-card poker hands.

• b. A heart flush is a 5-card hand that consists of all hearts. Find the number of possible heart flush.

• c. Find the probability of being dealt a heart flush?

• A. Total Combination:2,598,960

B. Possible Combination for heart flush: 1287

C. Probability: 1287/2,598,960

Probability with Combinations (Problem # 7)

• To play the Powerball, a player needs to select five numbers from 1-59 and a powerball number from 1-39.

a. What is the chance of winning the top prize if you buy one ticket?

b. What is the chance of winning the top prize if the cost for one ticket is $1 and you spent $100?

• A. 1/195,249,054• B. 50/97,624,527

Probability with Combinations (Problem # 8)

• To play the DC Lotto Hot Sizzler, a player must select 5 numbers from 1-39 and 1 number from 1-19. What is the probability of winning the top prize if you only purchased one ticket? 100 tickets? 1000 tickets? 10,000 tickets?

• One ticket: 1/10,939,383

• 1000 tickets: 1000/10,939,383

• 10,000 tickets:10,000/10,939,383

Probability with Combinations (Problem # 9)

• A group consists of 6 men and 10 women. Four members are selected at random to attend a conference. Find the probability that the selected group consists of

• A. All men• B. All women• C. 2 men and 2 women

• A. 3/364• B. 3/26• C. 135/364

Probability with Combinations (Problem # 10)

• A box contains 20 IPods, where 5 of them are defective. If 5 are selected at a random, find the probability that

• A. All are defective• B. None are defective

• A. 1/15504• B. 1001/5168