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16 Sep 2014, 00:54
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C
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59% (01:24) correct
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If a soccer match ended with a score of 3:2 and assuming that all possible scoring scenarios of the match had an equal probability, what is the probability that the team which ultimately lost had scored the first goal?
A. \(\frac{1}{4}\)
B. \(\frac{3}{10}\)
C. \(\frac{2}{5}\)
D. \(\frac{5}{12}\)
E. \(\frac{1}{2}\)
A. \(\frac{1}{4}\)
B. \(\frac{3}{10}\)
C. \(\frac{2}{5}\)
D. \(\frac{5}{12}\)
E. \(\frac{1}{2}\)
16 Sep 2014, 00:54
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Official Solution:
If a soccer match ended with a score of 3:2 and assuming that all possible scoring scenarios of the match had an equal probability, what is the probability that the team which ultimately lost had scored the first goal?
A. \(\frac{1}{4}\)
B. \(\frac{3}{10}\)
C. \(\frac{2}{5}\)
D. \(\frac{5}{12}\)
E. \(\frac{1}{2}\)
Let's start by determining the number of possible scoring scenarios for the match. We can represent a goal scored by the winning team with W and a goal scored by the losing team with L. The total number of possible scoring scenarios is the number of ways to arrange the letters (L, L, W, W, W), which is equal to \(\frac{5!}{2!3!}=10\).
To find the number of scenarios in which the losing team scored the first goal, we can fix L in the first position and arrange the remaining four letters (L, W, W, W) in \(\frac{4!}{3!}=4\) ways. Therefore, the probability that the team which ultimately lost had scored the first goal is \(P=\frac{favorable}{total}=\frac{4}{10}=\frac{2}{5}\).
Alternatively, we can consider the distribution of (L, L, W, W, W) into a grid representing the sequence of goals (first goal, second goal, third goal, fourth goal, fifth goal). What is the probability that we select an L when choosing which letter to put in the first goal position? Since there are two L's out of a total of five letters, the probability is \(P=\frac{2}{5}\).
Answer: C
If a soccer match ended with a score of 3:2 and assuming that all possible scoring scenarios of the match had an equal probability, what is the probability that the team which ultimately lost had scored the first goal?
A. \(\frac{1}{4}\)
B. \(\frac{3}{10}\)
C. \(\frac{2}{5}\)
D. \(\frac{5}{12}\)
E. \(\frac{1}{2}\)
Let's start by determining the number of possible scoring scenarios for the match. We can represent a goal scored by the winning team with W and a goal scored by the losing team with L. The total number of possible scoring scenarios is the number of ways to arrange the letters (L, L, W, W, W), which is equal to \(\frac{5!}{2!3!}=10\).
To find the number of scenarios in which the losing team scored the first goal, we can fix L in the first position and arrange the remaining four letters (L, W, W, W) in \(\frac{4!}{3!}=4\) ways. Therefore, the probability that the team which ultimately lost had scored the first goal is \(P=\frac{favorable}{total}=\frac{4}{10}=\frac{2}{5}\).
Alternatively, we can consider the distribution of (L, L, W, W, W) into a grid representing the sequence of goals (first goal, second goal, third goal, fourth goal, fifth goal). What is the probability that we select an L when choosing which letter to put in the first goal position? Since there are two L's out of a total of five letters, the probability is \(P=\frac{2}{5}\).
Answer: C
14 Aug 2016, 08:14
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Hi,
I could not understand why we have only two arrangements? I understand that there are only two Ls and occurring first can happen in two ways but what is conceptually wrong in considering these three options -LWWLW, LLWWW & LWLWW? In all previous options loser is scoring first.
Please explain.
Regards
Yash
I could not understand why we have only two arrangements? I understand that there are only two Ls and occurring first can happen in two ways but what is conceptually wrong in considering these three options -LWWLW, LLWWW & LWLWW? In all previous options loser is scoring first.
Please explain.
Regards
Yash
