The Question goes as follows:
Research has proven that 70% of the men has dark hair and that 25% of the women is blond. Furthermore it is known that 80% of the blonde women marries a dark-haired man.
Suppose that the number of men and women is equal, and that every man marries precisely one woman. Furthermore assume that every person has hair and that dark and blond are the only possible colours. Denote the hair colour of the man by X, and of the woman by Y .
(a) Determine the probability distributions of both X and Y (i.e. give the probability of each possible outcome).
(b) Determine the conditional probability distributions (X | Y = blond) and (X | Y = dark).
(c) Calculate H(X) in 3 decimals.
(d) Show that the uncertainty of the man’s hair colour becomes larger if it is known that his wife has dark hair. Round up to 3 decimals.
If i'd say + is having dark hair and - is having blond hair and <3 is marries
Then i have:
- P(X+) = 0.7
- P(X-) = 0.3
- P(Y+) = 0.75
- P(Y-) = 0.25
- P(Y-| X+) = P(Y- <3 X+) * P(Y-) = 0.8 * 0.25 = 0.2 (Not sure if this is correct) P(Y-| X-) = opposite of (P(Y- <3 X+)) * P(Y-) = 0.2 * 0.25 = 0.05 (Again not sure if this is correct)
Then I could solve P(X+|Y-) = (P(Y-| X+) *P(X+))/ P(Y-) = (0.2 * 0.25)/0.25 and P(X-| Y-) = (P(Y-|X-) * P(X-))/P(Y-) = (0.05 * 0.3)/0.25 =0.06 with Bayes Theorem.
But I am not sure how to continue and solve P(Y+ | X+) and therefore P(X+ | Y+) and so on.
Thanks in advance for a reply.
Best regards,
Paul