(a) Let {A1, A2} be a partition of a sample space and let B be any event. State and prove the Law of Total Probability as it applies to the partition {A1, A2} and the event B.
There are two coins in a box; one coin is a fair coin, and the other is a biased coin thatcomes up heads 75 percent of the time. Fiona chooses a coin at random and tosses it.
(b) What is the probability that the chosen coin flips heads? (Hint: The Law of TotalProbability.)
(c) Given that the chosen coin flips heads, what is the conditional probability that it wasthe biased coin? (Hint: Bayes’ Theorem.)
(d) Given that the chosen coin flips tails, what is the conditional probability that it wasthe biased coin?

Given that {A1, A2} be a partition of a sample space and let B be any event. State and prove the Law of Total Probability as it applies to the partition {A1, A2} and the event B.

Since A1 and A2 are mutually exclusive and exhaustive, we can say

b) P(B) = P(A1B)+P(A2B)

Selecting any one coin is having probability 0.50. and A1, A2 are events that the coins show heads.[tex]P(B/A1) = 0.50 \\P(B/A2) = 0.75\\P(A1B) = 0.5(0.5) = 0.25 \\P(A2B) = 0.75(0.5) = 0.375\\P(B) = 0.625[/tex]

c) Using Bayes theorem

conditional probability that it wasthe biased coin

=[tex]\frac{0.375}{0.625} \\=\frac{3}{5}[/tex]

d) Given that the chosen coin flips tails,the conditional probability that it was the biased coin=[tex]\frac{0.25*0.5}{0.25*0.5+0.5*0.5} \\=\frac{1}{3}[/tex]