BTGModeratorVI wrote: ↑

Fri Apr 10, 2020 8:12 am

A gambler rolls three fair six-sided dice. What is the probability that two of the dice show the same number, but the third shows a different number?

A) 1/8
B) 5/18
C) 1/3
D) 5/12
E) 5/6

Answer: D
Source: Manhattan prep

Let's first calculate P(same, same, different)
P(same, same, different) = P(1st roll is ANY value AND 2nd roll matches 1st roll AND 3rd roll is different from first 2 rolls)
= P(1st roll is ANY value) x P(2nd roll matches 1st roll) x P(3rd roll is different from first 2 rolls)
= 6/6 x 1/6 x 5/6
= 5/36
So, P(same, same, different) = 5/36

However, this is not the only way to get 2 sames and 1 different.
There's also: same, different, same as well as different, same, same

Applying similar logic, we know that P(same, different, same) = 5/36
And P(different, same, same) = 5/36

So, P(2 same rolls and 1 different) = P(same, same, different or different, same, same or same, different, same)
= P(same, same, different ) + P(different, same, same) + P(same, different, same)
= 5/36 + 5/36 + 5/36
= 15/36
= 5/12

Answer: D

Cheers,
Bret