Time for a Swim!

Consider the probability that the number of boys didn't exceed the number of girls when they jumped in.

In order for this to happen, the first person to get in must be a girl ( $\frac{1}{2}$ probability), and the last to get out must be a boy ( $\frac{3}{5}$ probability if the first person to get in was a girl). There are six possibilities for the 2nd through 5th to jump in:

• FFMM
• FMFM
• FMMF
• MFFM
• MFMF
• MMFF

Of, these all but the last one fit the condition that the number of men never exceeds the number of women, ( $\frac{5}{6}$ probability) assuming the first to jump in was a girl and the last was a boy.

So, the probability when they first get in that the number of boys doesn't exceed the number of girls is:

$P_{\text{getting in}} = \frac{1}{2}\cdot\frac{3}{5}\cdot\frac{5}{6} = \frac{1}{4}$

Similarly on the way out the probability is

$P_{\text{getting out}} = \frac{1}{4}$

So, the probability over all, is given by:

$P = P_{\text{getting in}}P_{\text{getting out}} = \frac{1}{4}\cdot\frac{1}{4} = \frac{1}{16} = \boxed{0.0625}$