# Expected value – number of boys

by Brian
(CA)

Compute Sample Space and Expected Value

What is the expected number of boys in a four-child family?

Determine the sample space, and show the probability distribution in a table.

### Comments for Expected value – number of boys

 Jul 15, 2013 Compute Expected Value by: Staff Answer Part I While the chance of a boy being born is slightly higher than the chance of a girl being born, those differences are very small. The chance of a girl or boy being born is almost the same. Therefore, for this problem we can assume the chance of a boy being born is 50% and the chance of a girl being born is also 50%. On this basis, the formula which you can use to calculate possible combinations of boys and girls in a four child family is: nCr = (n + r – 1)! / [(r!) ((n-1)!)] n = number of elements in the set we can choose from { boy, girl } = 2 r = number of elements selected (four children will be born) = 4 Order is not important Repetition is allowed nCr = (n + r – 1)! / [(r!) ((n-1)!)] 2C4 = (2 + 4 – 1)! / [(4!) ((2 - 1)!)] 2C4 = 5! / (4! * 1!) 2C4 = 5*4*3*2*1 / (4*3*2*1*1) 2C4 = 5/ 1 2C4 = 5 ------------------------------------------------------------------------

 Jul 15, 2013 Compute Expected Value by: Staff ------------------------------------------------------------------------ Part II There are only 5 possible boy-girl combinations when 4 children are born. The sample space is: {girl, girl, girl, girl} {boy, girl, girl, girl} {boy, boy, girl, girl} {boy, boy, boy, girl} {boy, boy, boy, boy} ------------------------------------------------------------------------

 Jul 15, 2013 Compute Expected Value by: Staff ------------------------------------------------------------------------ Part III Expected Value (boys): b = number of boys b0 = 0 boys b1 = 1 boy b2 = 2 boys b3 = 3 boys b3 = 4 boys P(E) = b0*P(b0) + b1*P(b1) + b2*P(b2) + b3*P(b3) + b4*P(b4) From the sample space we can construct a probability table for the number of boys Number of probability Boys, b P(b) 0 1/5 1 1/5 2 1/5 3 1/5 4 1/5 ------------------------------------------------------------------------

 Jul 15, 2013 Compute Expected Value by: Staff ------------------------------------------------------------------------ Part IV Expected Value: P(E) = 0 + 1/5 + 2/5 + 3/5 + 4/5 = 10/5 = 2 The expected number of boys in a four child family = 2 boys Thanks for writing. Staff www.solving-math-problems.com