Bunuel wrote:

Official Solution:

If a certain soccer game ended 3:2, what is the probability that the side that lost scored first? (Assume that all scoring scenarios have the same probabiliy)

A. \(\frac{1}{4}\)

B. \(\frac{3}{10}\)

C. \(\frac{2}{5}\)

D. \(\frac{5}{12}\)

E. \(\frac{1}{2}\)

Consider empty slots for 5 goals: *****. Say \(W\) is a goal scored by the winner and \(L\) is a goal scored by the loser. We need the probability of when \(L\) comes first while distributing these goals (5 letters \(LLWWW\)) into 5 slots.

Since there are 2 \(L\)'s out of total 5 letters, then \(P=\frac{Favorable}{Total}=\frac{2}{5}\).

Answer: C

Is the reason why this is not 1/2 because the potential outcomes have already been dictated by the score? In other words, if the question stem had not provided the score of the game and the question was "what is the probability that Team A scores first?", then that probability would be 1/2?

However, because you are given the final score, there are 10 possible ways the game occurred:

LLWWW

LWLWW

LWWLW

LWWWL

WWWLL

WWLWL

WWLWL

WWLLW

WLWWL

WLLWW

WLWLW

The first four outcomes are the only ways the team could have not gotten the first goal out of the ten outcomes. P = 4/10 = 2/5

Correct?

P.S. - Can someone please express this in a combinatoric approach? I understand that there are 5c2 ways to arrange LLWWW, but how do we get the numerator to be 4? I believe it's (2c1)*2, since order of "which" L doesn't matter, but can someone please confirm?