A person goes to the office either by a car, or scooter, or bus, the probability of which being `(2)/(7),(3)/(7),(2)/(7)` respectively. The probability that he reaches office late if he takes a car, or scooter, or bus is `(2)/(9),(1)/(9),(4)/(9)` respectively. If he reaches office in time, the probability that he travelled by car is k, then the value of `24k+7` is equal to

Options

A. 7
B. 14
C. `(21)/(2)`
D. `(31)/(2)`

Answer

To solve the problem step by step, we will calculate the required probabilities and then find the value of \(24k + 7\).

### Step 1: Define the probabilities
Let:
- \(P(C) = \frac{2}{7}\) (Probability of going by car)
- \(P(S) = \frac{3}{7}\) (Probability of going by scooter)
- \(P(B) = \frac{2}{7}\) (Probability of going by bus)

### Step 2: Define the probabilities of being late
- \(P(L|C) = \frac{2}{9}\) (Probability of being late given he took the car)
- \(P(L|S) = \frac{1}{9}\) (Probability of being late given he took the scooter)
- \(P(L|B) = \frac{4}{9}\) (Probability of being late given he took the bus)

### Step 3: Calculate the probabilities of being on time
Using the complement rule:
- \(P(L'|C) = 1 - P(L|C) = 1 - \frac{2}{9} = \frac{7}{9}\)
- \(P(L'|S) = 1 - P(L|S) = 1 - \frac{1}{9} = \frac{8}{9}\)
- \(P(L'|B) = 1 - P(L|B) = 1 - \frac{4}{9} = \frac{5}{9}\)

### Step 4: Calculate the total probability of reaching on time
Using the law of total probability:
\[ P(L') = P(C) \cdot P(L'|C) + P(S) \cdot P(L'|S) + P(B) \cdot P(L'|B) \]
Substituting the values:
\[ P(L') = \left(\frac{2}{7} \cdot \frac{7}{9}\right) + \left(\frac{3}{7} \cdot \frac{8}{9}\right) + \left(\frac{2}{7} \cdot \frac{5}{9}\right) \]
Calculating each term:
\[ = \frac{2 \cdot 7}{7 \cdot 9} + \frac{3 \cdot 8}{7 \cdot 9} + \frac{2 \cdot 5}{7 \cdot 9} \]
\[ = \frac{14}{63} + \frac{24}{63} + \frac{10}{63} \]
\[ = \frac{48}{63} \]

### Step 5: Calculate the probability of traveling by car given that he is on time
Using Bayes' theorem:
\[ P(C|L') = \frac{P(L'|C) \cdot P(C)}{P(L')} \]
Substituting the values:
\[ P(C|L') = \frac{\left(\frac{7}{9}\right) \cdot \left(\frac{2}{7}\right)}{\frac{48}{63}} \]
Calculating:
\[ = \frac{\frac{14}{63}}{\frac{48}{63}} = \frac{14}{48} = \frac{7}{24} \]
Thus, \(k = \frac{7}{24}\).

### Step 6: Calculate \(24k + 7\)
Substituting \(k\):
\[ 24k + 7 = 24 \cdot \frac{7}{24} + 7 = 7 + 7 = 14 \]

### Final Answer
The value of \(24k + 7\) is \(14\).

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