An expected value question

[Lazenby57]

Find the expected value for the number of girls for a family of three children.

I know the formula for the this (the probability of the first event ocurring times the net amount gained or lost plus the probability of the second event occurring times the amount gained or lost in that even and so on for total event numbers ) but I'm having trouble applying it for this problem.

 

[pka]

Birth of any gender is binomial. The expected value of a binomial random variable with probability p is E=np where n is the number of trials.

 

[soroban]

Hello, Lazenby57!

Find the expected value for the number of girls for a family of three children.

You can crank out all the possible cases ... there are eight outcomes.

. . \(\displaystyle \begin{array}{cccc}1) & G & G & G \\ 2) & G & G & B \\ 3) & G & B & G \\ 4) & G & B & B \\ 5) & B & G & G \\ 6) & G & G & B\\ 7) & B & B & G \\ 8) & B & B & B\end{array}\)

\(\displaystyle \begin{array}{ccc}\text{# girls } & \text{ outcomes } & \text{ probabiity} \\
3 & 1 & \frac{1}{8} \\
2 & 3 & \frac{3}{8} \\
1 & 3 & \frac{3}{8} \\
0 & 1 & \frac{1}{8}\end{array}\)


Therefore: \(\displaystyle \:E \:=\3)\left(\frac{1}{8}\right)\,+\,(2)\left(\frac{3}{8}\right) \,+\,(1)\left(\frac{3}{8}\right)\,+\,(0)\left(\frac{1}{8}\right) \:=\:\L\fbox{1\frac{1}{2}}\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This makes sense!

With any number of children, we'd expect half to be girls, right?

 

[stapel]

P.S. to Lazenby57: Welcome to FreeMathHelp!

Eliz.