Consider a random experiment where two fair coins are tossed. Let A be the…
2023
Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE?
- A.
A and B are independent.
- B.
A and C are independent.
- C.
B and C are independent.
- D.
Prob(B|C) = Prob(B)
Attempted by 65 students.
Show answer & explanation
Correct answer: C, D
Sample space and event probabilities:
Sample space: {HH, HT, TH, TT}, each outcome has probability 1/4.
Event A (HEAD on both throws): {HH}, P(A) = 1/4.
Event B (HEAD on first throw): {HH, HT}, P(B) = 1/2.
Event C (HEAD on second throw): {HH, TH}, P(C) = 1/2.
Check independence between each pair:
For the pair 'both HEAD' and 'HEAD on first throw': P(A∩B)=P({HH})=1/4, but P(A)P(B)=(1/4)*(1/2)=1/8. Not equal, so these events are not independent.
For the pair 'both HEAD' and 'HEAD on second throw': P(A∩C)=P({HH})=1/4, but P(A)P(C)=(1/4)*(1/2)=1/8. Not equal, so these events are not independent.
For the pair 'HEAD on first throw' and 'HEAD on second throw': P(B∩C)=P({HH})=1/4 and P(B)P(C)=(1/2)*(1/2)=1/4. Equal, so these events are independent.
Use independence to check the conditional probability statement:
P(B|C) = P(B∩C)/P(C) = (1/4)/(1/2) = 1/2, which equals P(B). So the equality P(B|C) = P(B) holds.
Final answer:
True: B and C are independent; Prob(B|C) = Prob(B).
False: A and B are independent; A and C are independent.