Out of 13 applicants for a job, there are 5 women and 8 men. It is desired to select 2 persons for the job. The probability that at least one of the selected persons will be a woman is

Options

A. `25//39`
B. `14//39`
C. `5//13`
D. `10//13`

Answer

To solve the problem of finding the probability that at least one of the selected persons for the job is a woman, we can follow these steps:

### Step 1: Determine the total number of applicants
We have a total of 13 applicants, consisting of 5 women and 8 men.

### Step 2: Calculate the total number of ways to select 2 persons from 13 applicants
The total number of ways to select 2 persons from 13 can be calculated using the combination formula:
\[ \text{Total ways} = \binom{13}{2} = \frac{13 \times 12}{2 \times 1} = 78 \]

### Step 3: Calculate the number of favorable outcomes (at least one woman)
To find the probability of selecting at least one woman, we can consider two cases:
1. Selecting 1 woman and 1 man.
2. Selecting 2 women.

#### Case 1: Selecting 1 woman and 1 man
- The number of ways to choose 1 woman from 5 is:
\[ \binom{5}{1} = 5 \]
- The number of ways to choose 1 man from 8 is:
\[ \binom{8}{1} = 8 \]
- Therefore, the total number of ways to select 1 woman and 1 man is:
\[ 5 \times 8 = 40 \]

#### Case 2: Selecting 2 women
- The number of ways to choose 2 women from 5 is:
\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 4: Calculate the total number of favorable outcomes
Now we can add the outcomes from both cases:
\[ \text{Total favorable outcomes} = 40 + 10 = 50 \]

### Step 5: Calculate the probability
The probability of selecting at least one woman is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{at least one woman}) = \frac{\text{Total favorable outcomes}}{\text{Total ways}} = \frac{50}{78} \]
This can be simplified:
\[ P(\text{at least one woman}) = \frac{25}{39} \]

### Final Answer
Thus, the probability that at least one of the selected persons will be a woman is:
\[ \frac{25}{39} \]

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