0
$\begingroup$

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:

dbinom(0:5,5,0.5)

0.03125

0.15625 

0.31250 

0.31250 

0.15625 

0.03125

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?

$\endgroup$
  • 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 $\endgroup$ – Michael Chernick Jun 4 '18 at 14:12
  • $\begingroup$ The results show a symmetry which should be the case when p=0.50. $\endgroup$ – Michael Chernick Jun 4 '18 at 14:13
1
$\begingroup$

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:

https://upload.wikimedia.org/wikipedia/commons/c/ca/Pascal_triangle_small.png

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

prop.table(c(1,5,10,10,5,1))
[1] 0.03125 0.15625 0.31250 0.31250 0.15625 0.03125
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.