Questions: For mutually exclusive events R1, R2, and R3, we have P(R1) = 0.05, P(R2) = 0.5, and P(R3) = 0.45. Also, P(Q R1) = 0.3, P(Q R2) = 0.7, and P(Q R3) = 0.8. Find P(R3 Q)

For mutually exclusive events R1, R2, and R3, we have P(R1) = 0.05, P(R2) = 0.5, and P(R3) = 0.45. Also, P(Q R1) = 0.3, P(Q R2) = 0.7, and P(Q R3) = 0.8. Find P(R3 Q)
Transcript text: For mutually exclusive events R_{1}, R_{2}, and R_{3}, we have P(R_{1}) = 0.05, P(R_{2}) = 0.5, and P(R_{3}) = 0.45. Also, P(Q | R_{1}) = 0.3, P(Q | R_{2}) = 0.7, and P(Q | R_{3}) = 0.8. Find P(R_{3} | Q)
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Solution

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Solution Steps

To find \( P(R_3 | Q) \), we can use Bayes' Theorem. The formula for Bayes' Theorem in this context is:

\[ P(R_3 | Q) = \frac{P(Q | R_3) \cdot P(R_3)}{P(Q)} \]

First, we need to calculate \( P(Q) \), which is the total probability of \( Q \) occurring. This can be found using the law of total probability:

\[ P(Q) = P(Q | R_1) \cdot P(R_1) + P(Q | R_2) \cdot P(R_2) + P(Q | R_3) \cdot P(R_3) \]

Once we have \( P(Q) \), we can substitute it back into Bayes' Theorem to find \( P(R_3 | Q) \).

Step 1: Calculate the Total Probability of \( Q \)

To find \( P(Q) \), we use the law of total probability:

\[ P(Q) = P(Q | R_1) \cdot P(R_1) + P(Q | R_2) \cdot P(R_2) + P(Q | R_3) \cdot P(R_3) \]

Substituting the given values:

\[ P(Q) = (0.3 \times 0.05) + (0.7 \times 0.5) + (0.8 \times 0.45) \]

\[ P(Q) = 0.015 + 0.35 + 0.36 = 0.725 \]

Step 2: Apply Bayes' Theorem to Find \( P(R_3 | Q) \)

Using Bayes' Theorem, we calculate \( P(R_3 | Q) \):

\[ P(R_3 | Q) = \frac{P(Q | R_3) \cdot P(R_3)}{P(Q)} \]

Substituting the known values:

\[ P(R_3 | Q) = \frac{0.8 \times 0.45}{0.725} \]

\[ P(R_3 | Q) = \frac{0.36}{0.725} \approx 0.4966 \]

Final Answer

The probability \( P(R_3 | Q) \) is approximately:

\[ \boxed{0.4966} \]

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