Probability using venn diagrams
Two students from the group are chosen at random. Given that x = 10, find the probability that both of these students each owns exactly two out of the three items [laptop, tablet, games machine].
My answer: I have found it by using P(L&G) or P(L&T) or P(G&T) = $\dfrac{10}{300}$ + $\dfrac{12}{300}$ + $\dfrac{15}{300}$ = $\dfrac{37}{300}$.
The correct answer is 0.0148.
May I know what is wrong with my answer? Thanks.
$\endgroup$2 Answers
$\begingroup$$\dfrac{10}{300}$ + $\dfrac{12}{300}$ + $\dfrac{15}{300}$ = $\dfrac{37}{300}$ is the probability that given a single student, they own exactly two out of the three items.
Now, given that the first student owns exactly two out of the three games, the probability that the second student also owns exactly two out of the three games is:
$\dfrac{36}{299}$
The $300$ becomes a $299$ because we do not recount the first student. The $37$ becomes a $36$ for essentially the same reason, we do not want to recount the first student, whom we know owns exactly $2$ out of the $3$ games.
Thus the probability that two students randomly selected both own exactly $2$ of the $3$ games is:
$(\frac{37}{300})(\frac{36}{299})=\frac{1332}{89700}=.0148494983$
Lets rephrase all of this using standard probability notation. Let $A$ be the event that the first student owns exactly two out of the three games and let $B$ be the event that the second student owns exactly two out of the three games. Then we want to calculate the probability:
$P(A)P(B|A)$
Where $P(B|A)$ denotes the conditional probability of $B$ given $A$, or rather, the chance that event $B$ will occur GIVEN that event $A$ will occur. So, as above we have that:
$P(A)=\frac{37}{300}$ and $P(B|A)=\frac{36}{299}$
$\endgroup$ 1 $\begingroup$The probability that you found is for choosing a single student. To account for the choosing the second student (assuming independence) you need to multiply by the probability that the second student has exactly two of the items. Note that in this second probability, the first student is no longer available as a choice from the sample space, so they are also no longer in the group of students with two of the items.
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