A coin is tossed thrice. Let X be the event that head occurs in each of the first two tosses. Let Y be the event that a tail occurs on the third toss. Let Z be the event that two tails occur in three tosses. Based on the above information, which one of the following statements is TRUE? A. X and Y are not independent B. Y and Z are not independent C. X and Z are dependent D. Y and Z are independent

GATE 2015 · General Aptitude · Conditional Probability · medium

Answer: Y and Z are NOT independent (P(Y and Z) = 1/4 but P(Y)*P(Z) = 3/16). Answer: B.

1. Find individual probabilities: P(X) = 2/8 = 1/4 (HHH, HHT). P(Y) = 4/8 = 1/2 (HHT, HTT, THT, TTT). P(Z) = 3/8 (HTT, THT, TTH).
2. Test X and Y for independence: X and Y = {HHT}. P(X and Y) = 1/8. P(X)*P(Y) = (1/4)*(1/2) = 1/8. Equal, so X and Y are INDEPENDENT.
3. Test Y and Z for independence: Y and Z = outcomes in both Y and Z = {HTT, THT} (exactly two tails with third toss = T). P(Y and Z) = 2/8 = 1/4. P(Y)*P(Z) = (1/2)*(3/8) = 3/16. Since 1/4 = 4/16 != 3/16, Y and Z are DEPENDENT (NOT independent).
4. Verify X and Z for completeness: X and Z = {HHH, HHT} intersect {HTT, THT, TTH} = {} (no outcome is in both). P(X and Z) = 0. P(X)*P(Z) = (1/4)*(3/8) = 3/32 != 0. So X and Z are also DEPENDENT.