I have probability questions and am confused about Bayes Theorem.

27- I thought P(A | B) = (1/10)/ (2/5)

P(A) = 1/10

P(B) = 2/5

P(A and B) = 1/10

but the answer was not correct

31- 33

31- I understand the answer P(HH | H) is 2/3, but I am unsure how to use the formula.

32- I do not understand the mutually exclusive meaning of this question.

If so, are A and B independent?

33- I understand A and B and C are independent, but I cannot solve P(C and (A U B))

Correct answers:

27 - 1/7

31- 2/3

32- 0.34

33- 7/30

Could you help me?

Thanks,

31. There are two coins in a hat. One coin has two heads. The other coin has one head and one tail. Shannon reaches into the hat and pulls out a coin. The side facing up is heads. What is the probability that the side facing down is a head also? 32. Let A,B,C be three events. You are given: P[A] = 0.3 P[B] = 0.2 P[C|A]=0.6 P[C|B]=0.8 You are also given that C A and CB are mutually exclusive. Calculate P[C n (A U 3)] . 33. Let A,B, C be three events. The events are pairwise independent but not mutually independent. The probability of each event is 1/3 and the probability that all three occur is 1/10. Calculate the probability that none of them occur. 27. A dresser drawer contains five gloves One pair of blue gloves, one pair of red gloves, and a single white glove. Suppose that Olivia is looking for two matching blue gloves. Olivia randomly simultaneously pulls two gloves out of the drawer without looking in the drawer. Let Event B be the event that at least one of the gloves is blue. Let Event A be the event that both gloves are blue. Calculate the conditional probability of Event A given that Event B occurs