SOLUTION: Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of a
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Question 270296: Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three boys in ten births.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
What you need is the probability of zero boys in 10 births plus the probability of one plus the probability of two plus the probability of 3.
The probability of exactly 
successes in 
trials given the probability of success on any given trial is 
and the probability of failure on any given trial is 
is given by:
\ =\ \left(n\cr k\right)p^kq^{n-k})
where )
is the number of ways to select things from things when order doesn't matter and is equal to !)
Probabilities are: 
because male and female are equally likely, and therefore 
.
And since you want "at most 3 boys" you need the sum:
\ =\ \sum_{k=0}^3\left(10\cr\,k\right)\left(0.5\right)^k\left(0.5\right)^{10-k})
Then for your first step, probability of exactly 3:
\ =\ \left(10\cr \,3\right)\left(0.5\right)^3\left(0.5\right)^7)
Then for exactly 2 you need:
\ =\ \left(10\cr \,2\right)\left(0.5\right)^2\left(0.5\right)^8)
And then repeat for exactly 1 and exactly 0. Finally add the 4 results.
John
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