3. Log-linear models

A study was conducted into the number of people with a given blood type between genders. Results were recorded for the number of people turning up for blood tests within a given day at a doctors' surgery and the corresponding contingency table is given below:

A B O

Male 12 17 9

female 7 15 12

Under the assumption that blood type and gender are independent, the number of people (count) within the (i, j)-th cell of this table, where i=1, 2, 3 denotes the blood types A, B and O, and j=1, 2 denotes the genders Male and Female, respectively, is Poisson distributed, determine the MLE estimate for the expected number of Females with blood type B and thus calculate the probability that exactly 20 people who turn up to the doctors surgery in a day will be Female with blood type B.