I have this question:

"There are a 100 families each with 5 children. Given that the null probability of having a boy is $p=0.5$, what is the probability of a family having 0,1,2,3,4,5 boys"

We have been asked specifically to use dbinom from R.

My solution:








My question:

Why are the numbers similar, as in why is the probability of having 0 boys the same as 5 boys, or why the probability of having 2 boys the same as having 3 boys?

  • 5
    $\begingroup$ Try writing out the math involved, i.e., the formula for the binomial distribution with $p = 0.5$ and $n = 5$, and it will become clear. $\endgroup$
    – jbowman
    Jun 4 '18 at 4:35
  • 1
    $\begingroup$ Hint: when there are two boys how many girls are there? $\endgroup$
    – mdewey
    Jun 4 '18 at 12:25
  • $\begingroup$ Shouldn't the probability of 3 girls and 2 boys be the same as having 3 boys and 2 girls since p=0.50 $\endgroup$ Jun 4 '18 at 14:12
  • $\begingroup$ The results show a symmetry which should be the case when p=0.50. $\endgroup$ Jun 4 '18 at 14:13

The Binomial coefficients have a symmetry which can be seen from their formula or visually from their relationship to Pascal's Triangle, as seen in this image from the wikipedia page:


Note that the dbinom output is just a normalised representation of the relevant row from Pascal's Triangle:

[1] 0.03125 0.15625 0.31250 0.31250 0.15625 0.03125

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