Question 481883: If a family has 3 children, find the probability that there is at least 1 boy among the children
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
This depends on what you consider the probability of a male child on any given trial. If you are just wanting to do an exercise in calculating a binomial distribution, then you can use 50% as the probabiltiy of a boy on any give trial. I reality, it doesn't actually work out that way. See:
Gender Odds
And note that the odds of a girl after having had one or two boys, (or vice versa) varies at each instance of a trial.
If you just need the binomial distribution on a 50/50 per trial basis, then you can either calculate the probability of exactly 1 boy, exactly 2 boys, and exactly 3 boys, then sum the three probabilities, or you can calculate the probability of exactly 3 girls (same as exactly 3 boys) and subtract from 1. That's because the only way you can't have at least 1 boy is to have all three girls.
The probability of  successes in  trials where  is the probability of success on any given trial is given by:
\ =\ \left(n\cr k\right\)\left(p\right)^k\left(1\,-\,p\right)^{n\,-\,k})
Where ) is the number of combinations of things taken at a time and is calculated by !})
For your problem,  ,  , and 
So:
\ =\ 1\ -\ \left(3\cr 3\right\)\left(0.5\right)^3\left(0.5\right)^{0})
Note that \ =\ 1\ \forall\ n\ >\ 0\ \in\ \mathbb{Z}) and 
Next time, flip a coin. You get the same 50/50 results, but each experiment is a good deal less messy and time consuming compared to having babies.
John
My calculator said it, I believe it, that settles it
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