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Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained.

Q. 12    Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).

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Two numbers are selected at random (without replacement) from the first six positive integers in 6\times 5=30 ways.

X denote the larger of the two numbers obtained.

X can be 2,3,4,5,6.

X=2, obsevations : (1,2),(2,1)

P(X=2)=\frac{2}{30}=\frac{1}{15}

X=3, obsevations : (1,3),(3,1),(2,3),(3,2)

P(X=3)=\frac{4}{30}=\frac{2}{15}

X=4, obsevations : (1,4),(4,1),(2,4),(4,2),(3,4),(4,3)

P(X=4)=\frac{6}{30}=\frac{3}{15}

X=5, obsevations : (1,5),(5,1),(2,5),(5,2),(3,5),(5,3),(4,5),(5,4)

P(X=5)=\frac{8}{30}=\frac{4}{15}

X=6, obsevations : (1,6),(6,1),(2,6),(6,2),(3,6),(6,3),(4,6),(6,4),(5,6),(6,5)

P(X=6)=\frac{10}{30}=\frac{1}{3}

Probability distribution is as follows:

       

X 2 3 4 5 6
P(X) \frac{1}{15} \frac{2}{15} \frac{3}{15} \frac{4}{15} \frac{1}{3}

 

E(X)=2\times \frac{1}{15}+3\times \frac{2}{15}+4\times \frac{3}{15}+5\times \frac{4}{15}+6\times \frac{1}{3}

 E(X)= \frac{2}{15}+ \frac{2}{5}+ \frac{4}{5}+ \frac{4}{3}+ \frac{2}{1}

E(X)= \frac{70}{15}

E(X)= \frac{14}{3}

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