Counting Probability Test Answer Key with Scoring – Precalculus

Question 1 Full Meals (2 points)

(a) 4 × 6 × 3 = 72.

(b) Skipping dessert adds one more dessert option: 4 × 6 × (3 + 1) = 96.

Answer: (a) 72; (b) 96

Scoring: 1 point for each part.

Question 2 Permutation or Combination (2 points)

P(7, 3) = 7 · 6 · 5 = 210. C(7, 3) = 2103! = 2106 = 35.

Permutations count each group in every possible order, so they are 3! = 6 times larger than combinations.

Answer: 210 and 35

Scoring: 1 point for each value; the explanation is part of the second point.

Question 3 Random Committee (3 points)

(a) C(10, 3) = 120.

(b) C(4, 2) · C(6, 1) = 6 · 6 = 36 committees, so P = 36120 = 310.

(c) Use the complement. No girls: C(6, 3) = 20. P(at least one girl) = 1 − 20120 = 56.

Answer: (a) 120; (b) 310; (c) 56

Scoring: 1 point for each part.

Question 4 Trophy Shelf (2 points)

(a) 5! = 120.

(b) The tallest trophy is fixed in the middle, and the other 4 can be arranged in 4! = 24 ways.

Answer: (a) 120; (b) 24

Scoring: 1 point for each part.

Question 5 Marbles Without Replacement (3 points)

(a) After a red draw, 4 red and 3 white remain among 7: 4/7 and 3/7. After a white draw, 5 red and 2 white remain among 7: 5/7 and 2/7.

(b) 58 · 47 = 2056 = 514.

(c) 58 · 37 + 38 · 57 = 1556 + 1556 = 3056 = 1528.

Answer: (b) 514; (c) 1528

Scoring: 1 point for each part.

Question 6 Expected Value of a Game (3 points)

(a) 0.4 + 0.3 + 0.2 + 0.1 = 1, and every probability is between 0 and 1.

(b) E(X) = (−2)(0.4) + 0(0.3) + 3(0.2) + 8(0.1) = −0.8 + 0 + 0.6 + 0.8 = 0.6.

(c) Over many plays the player gains about $0.60 per play on average.

Answer: E(X) = $0.60 per play

Scoring: 1 point for each part.

Question 7 Die Game (3 points)

Winnings: $12 with probability 16, $3 with probability 26, $0 with probability 36.

E(winnings) = 12 · 16 + 3 · 26 + 0 = 2 + 1 = $3.

Expected net gain = 3 − 5 = −$2. The player should expect to lose $2 per game.

Answer: expected winnings $3; expected net −$2; no

Scoring: 1 point for the distribution, 1 point for the expected winnings, 1 point for the net gain and conclusion.

Question 8 Pets Survey (2 points)

Divide each count by 40: P(0) = 0.20, P(1) = 0.40, P(2) = 0.25, P(3) = 0.15.

E = 0(0.20) + 1(0.40) + 2(0.25) + 3(0.15) = 0.40 + 0.50 + 0.45 = 1.35.

Answer: expected number of pets 1.35

Scoring: 1 point for the distribution, 1 point for the expected value.