Probability

John bets $1 that the next couple that arrives with three children will have two boys and one girl. Jim gives 3 to 1 odds. ( i.e. If John wins he collects $3 but if he loses he pays $1) What is John's expected gain or loss? (The probability of two boys and one girl is .375)

Hello, joash!

If you've been taught about Expected Value, the problem is straight forward.
If not, you should not be given this problem.
So where is your difficulty?

John bets $1 that the next couple that arrives with three children will have two boys and one girl.
Jim gives 3 to 1 odds. ( i.e. If John wins, he collects $3. But if he loses, he pays $1)
What is John's expected gain or loss?
(The probability of two boys and one girl is 0.375)

Click to expand...

\(\displaystyle P(\text{John wins \$3}) \:=\:\tfrac{3}{8}\)

\(\displaystyle P(\text{John loses \$1}) \:=\:\tfrac{5}{8}\)

\(\displaystyle E \;=\;\left(\tfrac{3}{8}\right)(+3) + \left(\tfrac{5}{8}\right)(-1) \;=\;\tfrac{9}{8} - \tfrac{5}{8} \:=\:\tfrac{4}{8} \:=\:+\tfrac{1}{2}\)

John can expect to win an average of 50 cents per game.