Question

Out of 13 applicants for a job, there are \(5 \mathrm{w}\) na 8 men, It is desired to select 2 persons for the job. The probability that atleast one of the selected persons will be a woman is: (a) \(\frac{25}{39}\) (b) \(\frac{31}{65}\) (c) \(\frac{25}{69}\) (d) \(\frac{5}{13}\)

Short answer

Answer: (a) \(\frac{25}{39}\)

Step-by-step solution

Total number of ways to select 2 people from 13

We can use the combination formula to find the total number of ways to select 2 people out of 13 applicants. The combination formula is given by: \(\mathrm{C(n, r)} = \frac{\mathrm{n!}}{\mathrm{r!(n-r)!}}\) Where n is the total number of applicants and r is the number of selected persons. In our case, n = 13 and r = 2. \(\mathrm{C(13, 2)} = \frac{\mathrm{13!}}{\mathrm{2!(13-2)!}}\)

Calculate the combinations

Now we calculate the combinations: \(\mathrm{C(13, 2)} = \frac{\mathrm{13!}}{\mathrm{2!(11)!}} = \frac{\mathrm{13\times12\times11!}}{\mathrm{2\times1\times11!}} = \frac{\mathrm{13\times12}}{\mathrm{2}} = 13\times6=78\)

Number of ways to select at least one woman

We can find the number of ways to select at least one woman by finding the complement of the event: the probability of selecting no woman (2 men). We will calculate the number of ways to select 2 men out of the 8 men using the combination formula. \(\mathrm{C(8, 2)} = \frac{\mathrm{8!}}{\mathrm{2!(8-2)!}}\)

Calculate the combinations for selecting 2 men

Now calculate the combinations for selecting 2 men: \(\mathrm{C(8, 2)} = \frac{\mathrm{8!}}{\mathrm{2!(6)!}} = \frac{\mathrm{8\times7\times6!}}{\mathrm{2\times1\times6!}} = \frac{\mathrm{8\times7}}{\mathrm{2}} = 4\times7=28\)

Calculate the number of ways to select at least one woman

To find the number of ways to select at least one woman, we subtract the number of ways to select 2 men from the total ways to select 2 people. Number of ways to select at least one woman = Total ways to select 2 people - Ways to select 2 men = 78 - 28 = 50

Calculate the probability

Now we can calculate the probability of selecting at least one woman by dividing the number of ways to select at least one woman by the total ways to select 2 people. Probability of selecting at least one woman = \(\frac{\text{Number of ways to select at least one woman}}{\text{Total ways to select 2 people}} = \frac{50}{78}=\frac{25}{39}\) So, the probability that at least one of the selected persons will be a woman is (a) \(\frac{25}{39}\).

Key concepts

Combination Formula

When solving probability problems involving the selection of items or people, it's essential to understand the combination formula. This formula is crucial for calculating the number of ways to choose a subset of items from a larger set, where the order of selection does not matter.

The combination formula is expressed as \(\mathrm{C(n, r)} = \frac{\mathrm{n!}}{\mathrm{r!(n-r)!}}\), where \(\mathrm{n!}\) (n factorial) is the product of all positive integers up to n, \(\mathrm{r}\) is the size of the subset, and \(\mathrm{(n-r)!}\) is the factorial of the difference between the total set size and the subset size.

An easy way to remember the combination formula is to think of it as the total number of ways to arrange \(\mathrm{r}\) items divided by the number of ways those \(\mathrm{r}\) items could be arranged amongst themselves, which is irrelevant when the order doesn't matter. For example, if you're selecting 2 people out of a group of 13, it's the same as asking, 'In how many ways can I pair people together, without caring who stands first or second?'.

Permutation and Combination

Understanding the difference between permutation and combination is vital for correctly solving various probability and counting problems. Permutations focus on sequences or arrangements where the order matters, while combinations deal with selections or groups where the order is irrelevant.

For instance, if we are looking at a password consisting of digits, the order of those digits is crucial (permutation); however, if we're selecting members for a committee, it usually doesn't matter who is chosen first or second (combination). An intuitive way to differentiate them is to ask yourself whether switching positions of items or people changes the outcome. If it does, you're dealing with a permutation. If not, you're working with combinations.

To avoid confusion, always remember that permutations are for lists (where order matters), and combinations are for groups (where order doesn't matter). This distinction simplifies the decision-making process when determining which mathematical principle to apply in various scenarios.

Probability Calculation

Probability calculation is a fundamental concept in statistics and mathematics, quantifying the chance that an event will occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Recalling our earlier example, when we calculate the probability of selecting at least one woman from a group of applicants, we first determine the total number of possible selections (all combinations of 2 people) and then identify the number of these combinations that meet our criteria (having at least one woman). The probability is then the ratio of these two numbers.

One key strategy in probability calculations is the use of the complement rule. It is sometimes easier to calculate the chance of the opposite (complementary) event and then subtract that probability from 1. The complementary rule is particularly handy when dealing with 'at least one' type problems, as it allows us to focus on the probability of the event not happening (in this case, selecting no women) and use it to find the desired probability (selecting at least one woman).

Probability concepts lay the foundation for many real-world applications such as predicting outcomes, assessing risks, and making informed decisions under uncertainty.