Question

Let \( X \) and \( Y \) be discrete random variables with \( P(Y \in \{0, 1\}) = 1 \), \[ P(X = 0) = \frac{3}{4}, \, P(X = 1) = \frac{1}{4}, \, P(Y = 1 | X = 1) = \frac{3}{4}, \, P(Y = 0 | X = 0) = \frac{7}{8}. \] Then \( 3P(Y = 1) - P(Y = 0) \) equals

Hint:

For conditional probabilities involving multiple events, use the law of total probability to break the problem into manageable parts.

Answer 1

Correct Answer: 0.12 - 0.13

Step 1: Find \( P(Y = 1) \).
We use the law of total probability to find \( P(Y = 1) \): \[ P(Y = 1) = P(Y = 1 | X = 1)P(X = 1) + P(Y = 1 | X = 0)P(X = 0). \] Substituting the given values: \[ P(Y = 1) = \frac{3}{4} \times \frac{1}{4} + P(Y = 1 | X = 0) \times \frac{3}{4}. \] From \( P(Y = 0 | X = 0) = \frac{7}{8} \), we find that \( P(Y = 1 | X = 0) = 1 - \frac{7}{8} = \frac{1}{8} \). So: \[ P(Y = 1) = \frac{3}{16} + \frac{3}{32} = \frac{9}{32}. \]
Step 2: Find \( P(Y = 0) \).
Since \( P(Y = 0) = 1 - P(Y = 1) \), we have: \[ P(Y = 0) = 1 - \frac{9}{32} = \frac{23}{32}. \]
Step 3: Compute \( 3P(Y = 1) - P(Y = 0) \).
Now, compute the desired expression: \[ 3P(Y = 1) - P(Y = 0) = 3 \times \frac{9}{32} - \frac{23}{32} = \frac{27}{32} - \frac{23}{32} = \frac{4}{32} = 0.125. \] Thus, the answer is approximately \( 0.12 \) to \( 0.13 \).

Step 4: Conclusion.
Thus, \( 3P(Y = 1) - P(Y = 0) = 0.12 \) to \( 0.13 \).