An observed event B can occure after one of the three events `A_(1), A_(2), A_(3)`. If `P(A_(1))= P_(A_(2))= 0.4, P(A_(3))=0.2` and `P(B//A_(1))= 0.25, P(B//A_(2))= 0.4, P(B//A_(3))= 0.125` what is the probability of `A_(1)` after observing B ?

Options

A. A. `(1)/(3)`
B. B. `(6)/(19)`
C. C. `(20)/(57)`
D. D. `(2)/(5)`

Answer

To find the probability of event \( A_1 \) after observing event \( B \), we can use Bayes' theorem. The formula for Bayes' theorem in this context is:

\[ P(A_1 | B) = \frac{P(B | A_1) \cdot P(A_1)}{P(B)} \]

### Step 1: Calculate \( P(B) \)

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

\[ P(B) = P(B | A_1) \cdot P(A_1) + P(B | A_2) \cdot P(A_2) + P(B | A_3) \cdot P(A_3) \]

Substituting the given values:

- \( P(A_1) = 0.4 \)
- \( P(A_2) = 0.4 \)
- \( P(A_3) = 0.2 \)
- \( P(B | A_1) = 0.25 \)
- \( P(B | A_2) = 0.4 \)
- \( P(B | A_3) = 0.125 \)

Now, we can calculate \( P(B) \):

\[ P(B) = (0.25 \cdot 0.4) + (0.4 \cdot 0.4) + (0.125 \cdot 0.2) \]

Calculating each term:

- \( 0.25 \cdot 0.4 = 0.1 \)
- \( 0.4 \cdot 0.4 = 0.16 \)
- \( 0.125 \cdot 0.2 = 0.025 \)

Now, adding these values together:

\[ P(B) = 0.1 + 0.16 + 0.025 = 0.285 \]

### Step 2: Calculate \( P(A_1 | B) \)

Now that we have \( P(B) \), we can substitute back into Bayes' theorem:

\[ P(A_1 | B) = \frac{P(B | A_1) \cdot P(A_1)}{P(B)} \]

Substituting the known values:

\[ P(A_1 | B) = \frac{0.25 \cdot 0.4}{0.285} \]

Calculating the numerator:

\[ 0.25 \cdot 0.4 = 0.1 \]

Thus, we have:

\[ P(A_1 | B) = \frac{0.1}{0.285} \]

### Step 3: Simplifying the Fraction

To simplify \( \frac{0.1}{0.285} \), we can multiply the numerator and the denominator by 1000 to eliminate the decimal:

\[ P(A_1 | B) = \frac{100}{285} \]

Now, we can simplify \( \frac{100}{285} \):

Dividing both the numerator and the denominator by 5:

\[ \frac{100 \div 5}{285 \div 5} = \frac{20}{57} \]

### Final Answer

Thus, the probability of \( A_1 \) after observing \( B \) is:

\[ P(A_1 | B) = \frac{20}{57} \]