There are two coins: Coin 1 is a fair coin that comes heads with probablity 1/2
ID: 3201334 • Letter: T
Question
There are two coins: Coin 1 is a fair coin that comes heads with probablity 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 E1 be the event that the first toss is heads, E2 be the event that the second toss is heads, and F be the event that the chosen coin is Coin 1.
(a) Are E1 and E2 independent? Justify your answer.
(b) Are E1 and E2 conditionally independent given F? Justify your answer.
Explanation / Answer
A) E1 and E2 are independent if the outcome of any of this will not affect the outcome of the other. E1 is the event that outcome of first toss is head and E2 is the event that the outcome of the second toss is head. The events doesnt have anything to do with the choosing of coins. The outcome of an event would have been affected only if the events involved chosing of any coins. So, both these events are independent of of each other.
B) In this case F is the event that the chosen coin is coin 1. So, dependent on F, the outcome of events E1 and E2 may vary since coin 2 is biased. So, unless F is given, the outcomes of E1 and E2 are not independent. When F is given, the outcome of E1 and E2 are conditionally independent as for the given value of F, the outcome of E1 and E2 wont vary with respect to the outcome of each other.