The ratio of number of officers and ladies in the Scorpion Squadron and in the Gunners Squadron are 3 : 1 and 2 : 5 respectively. An individual is selected to be the chairperson of their association. The chance that this individual is selected from the Scorpions is 2/3. Find the probability that the chairperson will be an officer.

Options

A. `25/42`
B. `13/43`
C. `11/43`
D. `7/42`

Answer

To solve the problem, we need to find the probability that the chairperson selected is an officer. We will use the given ratios and probabilities to calculate this step by step.

### Step 1: Determine the ratios of officers and ladies in each squadron.

For the Scorpion Squadron, the ratio of officers to ladies is given as 3:1. This means:
- Total parts = 3 (officers) + 1 (ladies) = 4 parts
- Probability of selecting an officer from Scorpion = \( \frac{3}{4} \)

For the Gunners Squadron, the ratio of officers to ladies is given as 2:5. This means:
- Total parts = 2 (officers) + 5 (ladies) = 7 parts
- Probability of selecting an officer from Gunners = \( \frac{2}{7} \)

### Step 2: Determine the probabilities of selecting from each squadron.

The probability of selecting an individual from the Scorpion Squadron is given as \( \frac{2}{3} \). Therefore, the probability of selecting from the Gunners Squadron is:
- Probability of selecting from Gunners = \( 1 - \frac{2}{3} = \frac{1}{3} \)

### Step 3: Calculate the overall probability of selecting an officer.

To find the total probability of selecting an officer, we can use the law of total probability:
\[ P(\text{Officer}) = P(\text{Officer | Scorpion}) \times P(\text{Scorpion}) + P(\text{Officer | Gunners}) \times P(\text{Gunners}) \]

Substituting the values we have:
\[ P(\text{Officer}) = \left(\frac{3}{4} \times \frac{2}{3}\right) + \left(\frac{2}{7} \times \frac{1}{3}\right) \]

### Step 4: Calculate each term.

First term:
\[ \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \]

Second term:
\[ \frac{2}{7} \times \frac{1}{3} = \frac{2}{21} \]

### Step 5: Combine the results.

Now, we need to add the two probabilities:
\[ P(\text{Officer}) = \frac{1}{2} + \frac{2}{21} \]

To add these fractions, we need a common denominator. The least common multiple of 2 and 21 is 42.

Convert \( \frac{1}{2} \) to have a denominator of 42:
\[ \frac{1}{2} = \frac{21}{42} \]

Now convert \( \frac{2}{21} \):
\[ \frac{2}{21} = \frac{4}{42} \]

Now we can add:
\[ P(\text{Officer}) = \frac{21}{42} + \frac{4}{42} = \frac{25}{42} \]

### Final Answer:
The probability that the chairperson will be an officer is \( \frac{25}{42} \).

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