There are four boxes `B_(1), B_(2), B_(3) and B_(4)`. Box `B_(i)` has `i` cards and on each card a number is printed, the numbers are from 1 to `i`. A box is selected randomly, the probability of selecting box `B_(i)` is `(i)/(10)` and then a card is drawn.
Let `E_(i)` respresent the event that a card with number 'i' is drawn, Then :
Q. `P(B_(3)|E_(2))` is equal to :

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(3)`
D. `(2)/(3)`

Answer

To solve the problem, we need to find the conditional probability \( P(B_3 | E_2) \), where \( B_i \) is the event of selecting box \( B_i \) and \( E_i \) is the event of drawing a card with the number \( i \).

### Step 1: Determine the probabilities of selecting each box
The probability of selecting box \( B_i \) is given by:
\[ P(B_i) = \frac{i}{10} \]
Thus, we have:
- \( P(B_1) = \frac{1}{10} \)
- \( P(B_2) = \frac{2}{10} = \frac{1}{5} \)
- \( P(B_3) = \frac{3}{10} \)
- \( P(B_4) = \frac{4}{10} = \frac{2}{5} \)

### Step 2: Calculate the probability of drawing a card with number 2, \( P(E_2) \)
To find \( P(E_2) \), we consider the cases when a card with the number 2 can be drawn:
- From box \( B_1 \): No card with number 2, so \( P(E_2 | B_1) = 0 \)
- From box \( B_2 \): There are 2 cards (1 and 2), so \( P(E_2 | B_2) = \frac{1}{2} \)
- From box \( B_3 \): There are 3 cards (1, 2, and 3), so \( P(E_2 | B_3) = \frac{1}{3} \)
- From box \( B_4 \): There are 4 cards (1, 2, 3, and 4), so \( P(E_2 | B_4) = \frac{1}{4} \)

Now we can calculate \( P(E_2) \) using the law of total probability:
\[ P(E_2) = P(E_2 | B_1)P(B_1) + P(E_2 | B_2)P(B_2) + P(E_2 | B_3)P(B_3) + P(E_2 | B_4)P(B_4) \]
Substituting the values:
\[ P(E_2) = 0 \cdot \frac{1}{10} + \frac{1}{2} \cdot \frac{2}{10} + \frac{1}{3} \cdot \frac{3}{10} + \frac{1}{4} \cdot \frac{4}{10} \]
Calculating each term:
\[ P(E_2) = 0 + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} \]

### Step 3: Calculate \( P(E_2 | B_3) \)
From box \( B_3 \), the probability of drawing card number 2 is:
\[ P(E_2 | B_3) = \frac{1}{3} \]

### Step 4: Apply Bayes' theorem to find \( P(B_3 | E_2) \)
Using Bayes' theorem:
\[ P(B_3 | E_2) = \frac{P(E_2 | B_3) P(B_3)}{P(E_2)} \]
Substituting the known values:
\[ P(B_3 | E_2) = \frac{\frac{1}{3} \cdot \frac{3}{10}}{\frac{3}{10}} = \frac{1}{3} \]

### Final Answer
Thus, the conditional probability \( P(B_3 | E_2) \) is:
\[ \boxed{\frac{1}{3}} \]