There are two coins: Coin 1 is a fair coin that comes heads with probability 1/2
ID: 2079202 • Letter: T
Question
There are two coins: Coin 1 is a fair coin that comes heads with probability 1/2 and tails with probability 1/2. Coin 2 is a fake coin that comes heads all the time. The following experiment is performed: We choose one of the two coins (Coin 1 or Coin 2) uniformly at random (each coin is chosen with probability 1/2), and consider two independent tosses of the chosen coin. Let E_1 be the event that the first toss is heads, E_2 be the event that the second toss is heads, and F be the event that the chosen coin is Coin 1. Are E_1 and E_2 independent? Justify your answer. Are E_1 and E_2 conditionally independent given F? Justify your answer.Explanation / Answer
Two events E1 and E2 are independent, if the fact that E1 occurs does not affect the probability of E2 occurring.
The test for independent events is that P(E1 E2) = P(E1). P(E2)
a) Assume fair coin is chosen,
Therefore, the probability of E1 occurring is, P(E1) = 0.5.
Since a fair coin is chosen, the probability of E2 occurring is, P(E2) = 0.5.
E1 E2 : The first toss is heads, second toss is heads; P(E1 E2) = P {(HH),(HT),(TH),(TT)} = ¼ = 0.25.
P(E1 E2) = P(E1). P(E2)
0.25 = 0.5*0.5
Assume unfair coin is chosen, where each toss results in HEADS.
The probability of E1 occurring is, P(E1) = 1.
Since a fair coin is chosen, the probability of E2 occurring is, P(E2) = 1.
E1 E2 : The first toss is heads, second toss is heads; P(E1 E2) = P {(HH)} = 1.
P(E1 E2) = P(E1). P(E2)
1 = 1.
b) The fact that F occurs meaning a fair coin is chosen.
Therefore, the probability of E1 occurring is, P(E1) = 0.5.
Since a fair coin is chosen, the probability of E2 occurring is, P(E2) = 0.5.
E1 E2 : The first toss is heads, second toss is heads; P(E1 E2) = P{(HH),(HT),(TH),(TT)} = ¼ = 0.25.
P(E1 E2) = P(E1). P(E2)
0.25 = 0.5*0.5