Suppose ππ for π = 1,2,3 are independent and identically distributed randomβ¦
2015
Suppose ππ for π = 1,2,3 are independent and identically distributed random variables whose probability mass functions are Pr[ππ = 0] = Pr[ππΒ = 1] = 1/2 for π = 1, 2, 3. Define another random variable π = π1π2 β π3, where β denotes XOR.
Then Pr[π = 0|π3 = 0] =_______________.
Answer: 0.75 β Key idea: given X3 = 0, the XOR with 0 leaves the other term unchanged, so Y = X1 X2. We want P(Y = 0 | X3 = 0) = P(X1 X2 = 0). X1 X2 equals 1 only when bothβ¦
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Correct answer: 0.75
Key idea: given X3 = 0, the XOR with 0 leaves the other term unchanged, so Y = X1 X2.
We want P(Y = 0 | X3 = 0) = P(X1 X2 = 0).
X1 X2 equals 1 only when both X1 = 1 and X2 = 1. Since the Xi are independent with P(Xi = 1) = 1/2,
P(X1 = 1 and X2 = 1) = (1/2) * (1/2) = 1/4.
Therefore P(X1 X2 = 0) = 1 - 1/4 = 3/4.
Answer: 3/4