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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

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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


Combinations formula:  

1. order is not important

2. 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



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Jul 15, 2013
Compute Expected Value
by: Staff


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Part II


Combinations formula for n = 2 and r = 4:  

1. order is not important

2. repetition is allowed





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}


Sample Space of possible number of boys in a four-child family






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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)


Expected value formula for number of boys in a family




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


Probability table:  number of boys in a four child family






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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



Expected Value Calculation:  number of boys in a four child family





Expected Value (final answer):  number of boys in a four child family








Thanks for writing.

Staff
www.solving-math-problems.com



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