Bluebook SAT Test Probability and Conditional Probability 5

Question 1

Events A and B satisfy P(A) = 0.6, P(B) = 0.5, and P(A ∪ B) = 0.8. Which statement is TRUE?

Accepted Answer

A and B are independent but not mutually exclusive

Answer

Use inclusion-exclusion: P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.6 + 0.5 − 0.8 = 0.3. Since P(A ∩ B) = 0.3 ≠ 0, A and B are NOT mutually exclusive. Check independence: P(A)·P(B) = 0.6 × 0.5 = 0.30 = P(A ∩ B). The condition holds, so A and B ARE independent. Answer: independent but not mutually exclusive.

Question 2

A factory has two machines. Machine X produces 60% of all items with a 4% defect rate; Machine Y produces 40% with a 2% defect rate. A randomly selected item is defective. What is the probability it came from Machine Y?

Accepted Answer

0.25

Answer

By Bayes' Theorem: P(defective) = (0.04)(0.60) + (0.02)(0.40) = 0.024 + 0.008 = 0.032. P(Y | defective) = (0.02 × 0.40) / 0.032 = 0.008 / 0.032 = 0.25. Despite Machine Y's lower defect rate, it accounts for only 25% of defective items because it produces fewer items overall.

Question 3

A standard 52-card deck is randomly split into two equal piles of 26. What is the probability the ace of spades and the ace of hearts end up in the same pile?

Accepted Answer

25/51

Answer

Fix the ace of spades in whichever pile it lands. Of the 51 remaining cards, exactly 25 slots remain in that same pile. The probability the ace of hearts occupies one of those 25 slots is 25/51 ≈ 0.490 — slightly less than 1/2. The common wrong answer is 1/2, but because one spot in the pile is already taken by the first ace, only 25 of the 51 remaining positions are in the same pile.

Question 4

P(A) = 0.4 and P(B | A) = 0.5. If A and B are independent, what is P(B | Aᶜ)?

Accepted Answer

0.5

Answer

Independence means P(B | A) = P(B). Since P(B | A) = 0.5, we get P(B) = 0.5. Independence also means P(B | Aᶜ) = P(B) = 0.5 — knowing whether A occurred (or did not) gives zero information about B. Both P(B | A) and P(B | Aᶜ) equal the unconditional P(B) when events are independent.

Question 5

A box contains 3 red and 2 blue balls. Two balls are drawn WITH replacement. Given that at least one ball is red, what is the probability that both balls are red?

Accepted Answer

3/7

Answer

P(both red) = (3/5)² = 9/25. P(at least one red) = 1 − P(both blue) = 1 − (2/5)² = 1 − 4/25 = 21/25. By conditional probability: P(both red | at least one red) = P(both red) / P(at least one red) = (9/25) ÷ (21/25) = 9/21 = 3/7. The trap answer 9/25 is the unconditional probability of both being red — not adjusted for the given condition.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Probability and Conditional Probability 5