1974 AMC 12 Problem 24

Attempt Problem 24 of the 1974 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1974 AMC 12 solutions, or check the answer key.

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24.

A fair die is rolled six times. The probability of rolling at least a five at least five times is

\frac{13}{729}

\frac{12}{729}

\frac{2}{729}

\frac{3}{729}

none of these

A roll is a success with probability

\frac{2}{6}=\frac{1}{3}

Add the disjoint cases of exactly five successes and exactly six successes

Let a success be a

5

6,

so its probability is

\frac{1}{3}.

The desired probability is

\begin{aligned} &\binom65\left(\frac13\right)^5 \left(\frac23\right) +\left(\frac13\right)^6\\ &\qquad=\frac{12}{729}+\frac1{729}\\ &\qquad=\frac{13}{729}. \end{aligned}

Therefore, the correct answer is A.

Problem 24 in other years