Question

Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?

$\dfrac{1}{64}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

$\dfrac{5}{16}$

$\dfrac{1}{2}$

Answer: D

Concepts:basic probability combinations

Difficulty rating: 900

Solution:

Note that for this restriction to hold, there must be $3$ balls of each color.

There are $2^6 = 64$ ways to color the balls and $\binom{6}{3} = 20$ to choose which balls are white.

The desired probability is therefore $\dfrac{20}{64} = \dfrac{5}{16}.$

Thus, D is the correct answer.

2021 AMC 10A Fall Problem 13

Answer: D

Difficulty rating: 900

Solution:

Problem 13 in Other Years