Question

Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?

$32$

$40$

$48$

$56$

$64$

Answer: D

Concepts:expected value complementary probability

Difficulty rating: 1070

Solution:

A coin ends as tails if and only if it has $3$ flips that are tails, which happens with probability $\frac 18.$ Thus, the probability of any coin being heads is $\frac 78 .$

Since each of the $64$ coins ends as heads with probability $\frac78,$ linearity of expectation gives the expected number of coins that are now heads: $64 \cdot \dfrac 78 = 56.$

Thus, the correct answer is D .

2015 AMC 10B Problem 18

Answer: D

Difficulty rating: 1070

Solution:

Problem 18 in Other Years