Problem 38 · AMC 8 Stretch
Stretch
Counting & Probability
binomial-probabilityor-process-add
Deanna guesses on a 4-question multiple-choice quiz. Each question has 4 choices, so each guess is correct with probability \(\tfrac{1}{4}\). You need at least 3 correct to pass. What is the probability she passes by guessing?
Show answer
Answer: 13/256 (about 0.05)
Show hints
Hint 1 of 4
Each question is a success (correct guess) with probability \(1/4\), or a failure with probability \(3/4\). Questions are independent.
Still stuck? Show hint 2 →
Hint 2 of 4
'At least 3 correct' out of 4 means exactly 3 OR exactly 4 — two separate cases to add (OR process).
Still stuck? Show hint 3 →
Hint 3 of 4
Exactly 4: only 1 pattern, \((1/4)^4\). Exactly 3: there are 4 patterns (which one is wrong), each \((1/4)^3 (3/4)\).
Show solution
Approach: Binomial probability — add the 'exactly 3' and 'exactly 4' cases
- Each guess is correct with probability \(p = \frac{1}{4}\) and wrong with \(q = \frac{3}{4}\). Passing needs at least 3 of 4 correct, so add 'exactly 3' and 'exactly 4' (OR). Use the common denominator \(4^4 = 256\).
- Exactly 4 correct (1 pattern): \(\left(\frac{1}{4}\right)^4 = \frac{1}{256}\).
- Exactly 3 correct (4 patterns — which question is wrong): \(4 \times \left(\frac{1}{4}\right)^3 \times \frac{3}{4} = 4 \times \frac{1}{64} \times \frac{3}{4} = \frac{12}{256}\).
- Add: \(\frac{12}{256} + \frac{1}{256} = \frac{13}{256} \approx 0.05\). Only about a 5% chance — studying is a much better plan!
Mark:
· log in to save