2025 AMC 10A Problem 16

Problem 16 of 25IntermediateCounting & Probability

There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?

Show solution

Solution

There are

3^3 = 27

equally likely placements. Of these,

3

pile all coins into one jar (max

3

), and

6

put one coin in each jar (max

1

). The other

18

split

2

1

(max

2

). So the expected maximum is

\frac{3 \cdot 3 + 2 \cdot 18 + 1 \cdot 6}{27} = \frac{51}{27} = \frac{17}{9}.

Therefore, the answer is D.

Answer choices

Solution

More practice