A certain Mr. Claus has mixed up his Scandinavian gift bags this Christmas, and he does not know which one goes to Denmark, which one goes to Sweden, and which one goes to Norway. The Danish bag has 70% soft gifts, the Norwegian one has 40%, whereas the Swedish bag has 20%.

Mr. Claus has now put one of the bags in his sleigh, and it’s your job as an engineering elf to find the probabilities of which country it is from. You sample 8 gifts, and by squeezing carefully, you determine that 4 of them are soft.

(g) You sample 42more gifts, to get a sharper guess beforeMr. Claus takes off. Seventeen of the gifts are soft. What are your new probabilities for which bag is on the sleigh?

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(a) Identify the alternatives Ak. (b) What are the values P(Ak)? (c) Identify the pivotal event B. (d) Calculate P(B|A). Note: the bags contain millions of gifts each, so the difference between sampling with and without replacement is negligible. (e) Calculate the posterior probabilities P(A|B). (f) Say in your own words what you just found out.