Probability of Having a Girl after Five Straight Boys

Date: 04/19/2005 at 15:16:24
From: Jenny
Subject: Odds of having a girl child

In a family with 5 children, ALL of whom are boys, what is the 
mathematical probability that the 6th child will be a girl?  Or
another boy?

I find it hard to believe that what I believed to be a 50/50 shot for
each pregnancy became a 100% male baby reality!  I am a desperate mom
needing either assurance that a girl IS mathematically possible, or
that it isn't looking good.  Thank you!

- Dreaming of Pink

Date: 04/22/2005 at 11:48:12
From: Doctor Wilko
Subject: Re: Odds of having a girl child

Hi Jenny,

Thanks for writing to Dr. Math!

The probability that your next child is a girl is 50%.  Likewise, the
probability that you have a boy is also 50%.

Actually, this isn't entirely accurate.  The true probability is not 
exactly 50/50 due to rates of conception, miscarriages, and other 
environmental/physiological factors.  In fact, one report I read said 
a boy is more likely to be born (with a probability of 51.21%) than a 
girl (with a probability of 48.79%).

However, to answer questions like yours:

  1. We assume that boys and girls are equally likely AND 

  2. We assume that having a boy or girl previously has no influence
     on the gender of the next child, i.e., we assume that births are
     independent events. 

Once we assume the outcomes (boy/girl) are equally likely and that 
whatever happens prior has no effect on the future outcomes, then we 
can model the situation with coin flipping.  

That is, in our simplified model, having a child is like flipping a
coin: you can get heads (boys) or tails (girls), and the previous coin
flips (genders of previous children) don't influence the next coin
flip (gender of the next child).  

Modeling the situation as a coin flip is why we often say that you 
have a 50% chance of having either a boy or girl. 
 
So hang in there, you still may have your girl!

Now here's a slightly different slant on the same question, which will
help illustrate how unlikely the overall scenario of six boys in a row
is.  Suppose before having your first child, you said, "We're going to
have six kids--what's the probability that all six will be boys?"

Since the probability of each child being a boy is 1/2, the overall
probability of six straight boys is:

  (1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2) = 1/64 = 0.015625 = 1.5625%

My point is that although the gender of each successive child is a 
50/50 chance, having a string of six boys (or girls) out of the 64
total possible boy/girl combinations is pretty rare!

Does this help?  Please write back if you need anything else.

- Doctor Wilko, The Math Forum
  http://mathforum.org/dr.math/