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  • Answer please! Two numbers are selected at random (without replacement) from the first six positive integers. If X denotes the smaller of the two numbers, then the expectation of X, is :

 Two numbers are selected at random (without replacement) from the first six positive integers.  If X denotes the smaller of the two numbers, then the expectation of X, is :

 

  • Option 1)  (5/3) 
  • Option 2) (14/3) 
  • Option 3) (13/3) 
  • Option 4) (7/3) 
 

Answers (1)

best_answer

Set of integers={1,2,3,4,5,6} = 30 

Number of ways to select two from set of integers = \small \binom{6}{2}=30 different ways
Out of this, 2 numbers are selected at random and let X denote the larger of the two numbers.

Since X is the large of the two numbers, X can assume the value of 2, 3, 4, 5 or 6.

P (X =2) = P (larger number is 2) = {(1,2) and (2,1)} = \small \frac{2}{30}

P (X = 3) = P (larger number is 3) = {(1,3), (3,1), (2,3), (3,2)} = \small \frac{4}{30}

P (X = 4) = P (larger number is 4) = {(1,4), (4,1), (2,4), (4,2), (3,4), (4,3)} = \small \frac{6}{30}

P (X = 5) = P (larger number is 5) = {(1,5), (51,), (2,5), (5,2), (3,5), (5,3), (4,5), (5.4)} = \small \frac{8}{30}

P (X = 6) = P (larger number is 6) = {(1,6), (6,1), (2,6), (6,2), (3,6), (6,3), (4,6), (6,4), (5,6), (6,5)} = \small \frac{10}{30}

Given the above probability distribution, the expected value or the mean can be calculated as follows:

\small \sum \left ( X_{i} \,\,\,\,x \,\,\,\,\,P\left ( X_{i} \right ) \right )=\\2\,\,x\,\,\frac{2}{30} + 3\times \frac{4}{30}+4\times \frac{6}{30} +5\times \frac{8}{30}+6\times \frac{10}{30}=\frac{4+12+24+40+60}{30}=\frac{14}{3}

 

 

Posted by

Anurag Uikey

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