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June 1st, 2017, 05:44 PM   #1
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Probability2

Three students are selected at random from a class of $10$ students among which $4$ students know $C$ programming of whom $2$ students are experts. If every such selection is equally likely, then the probability of selecting three students such that at least $2$ of them know $C$ programming with at least one out of the two selected being an expert in $C$ programming is

A)less than $1/4$
B)Greater than $1/4$ but less than $1/2$
C)Greater than $1/2$ but less than $3/4$
D)Greater than $3/4$

I got the answer $1/2$ after calculation, but there seems to be no exact option for it. Please check.

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June 1st, 2017, 06:05 PM   #2
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Quote:
Originally Posted by Lalitha183 View Post
Three students are selected at random from a class of $10$ students among which $4$ students know $C$ programming of whom $2$ students are experts. If every such selection is equally likely, then the probability of selecting three students such that at least $2$ of them know $C$ programming with at least one out of the two selected being an expert in $C$ programming is

A)less than $1/4$
B)Greater than $1/4$ but less than $1/2$
C)Greater than $1/2$ but less than $3/4$
D)Greater than $3/4$

I got the answer $1/2$ after calculation, but there seems to be no exact option for it. Please check.

Thank you
Let non-programmers be denoted by N, programmers by P, experts by E.

Valid combinations are

NPE, NEE, PEE, PPE

so the probability of any of these combos is

$\dfrac{
\binom{6}{1}\binom{2}{1}\binom{2}{1}+
\binom{6}{1}\binom{2}{0}\binom{2}{2}+
\binom{6}{0}\binom{2}{1}\binom{2}{2}+
\binom{6}{0}\binom{2}{2}\binom{2}{1}}{\binom{10}{3 }} = \dfrac{17}{60} $

which corresponds to answer (B)
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