Probability

[joash]

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)

 

[soroban]

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)

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

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

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

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