Problem 21 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A man and his wife appear for an interview for two posts. The probability of the husband's selection is \(1 / 7\) and that of the wife's selection is \(1 / 5\). What is the probability that only one of them be selected? (a) \(1 / 7\) (b) \(2 / 7\) (c) \(3 / 7\) (d) None of these

Step-by-Step Solution

Answer

The probability that only one of them gets selected is \( \frac{2}{7} \).

Step 1: Identify Probabilities of Individual Selections

Let's identify the probabilities of each individual being selected. The probability of the husband being selected is given as \( \frac{1}{7} \) and the probability of the wife being selected is \( \frac{1}{5} \).

Step 2: Calculate Probability of Both Being Selected

To find the probability that both are selected, multiply their individual probabilities: \( \frac{1}{7} \times \frac{1}{5} = \frac{1}{35} \).

Step 3: Calculate Probability of Neither Being Selected

Find the probability that neither gets selected by calculating the complement of each being selected first. The probability of the husband not being selected is \( 1 - \frac{1}{7} = \frac{6}{7} \). The probability of the wife not being selected is \( 1 - \frac{1}{5} = \frac{4}{5} \). Multiply these two probabilities: \( \frac{6}{7} \times \frac{4}{5} = \frac{24}{35} \).

Step 4: Calculate Probability of Only One Being Selected

To have only one selected, it could be either the husband or the wife. Calculate each case separately: 1. Only the husband is selected: \( \frac{1}{7} \times (1 - \frac{1}{5}) = \frac{1}{7} \times \frac{4}{5} = \frac{4}{35} \).2. Only the wife is selected: \( (1-\frac{1}{7}) \times \frac{1}{5} = \frac{6}{7} \times \frac{1}{5} = \frac{6}{35} \).Finally, add these probabilities for either husband or wife being selected: \( \frac{4}{35} + \frac{6}{35} = \frac{10}{35} = \frac{2}{7} \).

Step 5: Choose the Correct Answer from Given Options

From the calculations, the probability that only one of them gets selected is \( \frac{2}{7} \). The correct option from the given list that matches this probability is **(b) \( \frac{2}{7} \)**.

Key Concepts

Individual Probability

Individual probability refers to the likelihood of a single event occurring independently of any other events. In this scenario, we have two separate events: the husband getting selected and the wife being selected for the interview. The probability of each of these events is calculated independently.
1. **Probability of Husband's Selection:** This is given as \( \frac{1}{7} \). It means that out of every 7 similar events, the husband would be expected to be selected in 1 of them. 2. **Probability of Wife's Selection:** Similarly, the wife's individual probability of selection is \( \frac{1}{5} \). This indicates that out of 5 such opportunities, she might be selected once. Knowing the individual probabilities is crucial because they serve as the building blocks for calculating the probabilities of other outcomes, such as both being selected or only one being selected.

• Always focus on understanding each event in isolation before combining probabilities.
• Continuously verify the accuracy of these probabilities as they shape further calculations.

Complementary Probability

Complementary probability is about understanding the probability of an event **not** happening. In probability, each event has a complement which represents the occurrence of everything else but that event.
For our interview scenario:

• **Probability of Husband NOT Being Selected:** The complement of the probability that the husband is selected is calculated as:\[1 - \frac{1}{7} = \frac{6}{7}\]This means there is a \(\frac{6}{7}\) chance that the husband won't be selected.
• **Probability of Wife NOT Being Selected:** Similarly, for the wife,\[1 - \frac{1}{5} = \frac{4}{5}\]So, there is a \(\frac{4}{5}\) probability that the wife won't be selected.

Knowing these complementary probabilities helps in scenarios where we want to find out the likelihood of at least one event happening, neither happening, or other related outcomes.

Probability of Independent Events

When calculating the probability of independent events happening together, it's crucial to understand that the outcome of one event doesn't affect the other. This concept is essential in working out complex probabilities.
In this exercise, both the husband's and wife's selection are independent events. Therefore, to find out shared probabilities, we multiply individual probabilities.
For instance, to find the probability of both being selected, we calculate:\[\frac{1}{7} \times \frac{1}{5} = \frac{1}{35}\]Further, to determine the probability of **only one** being selected, consider each possible scenario:

• **Only Husband Selected:** This occurs if the husband is selected and the wife is not:\[\frac{1}{7} \times \frac{4}{5} = \frac{4}{35}\]
• **Only Wife Selected:** The wife gets selected, but the husband does not:\[\frac{6}{7} \times \frac{1}{5} = \frac{6}{35}\]

Add these probabilities to find the chance of only one being selected:\[\frac{4}{35} + \frac{6}{35} = \frac{10}{35} = \frac{2}{7}\]Understanding independent events is key in ensuring calculations are accurate when combining probabilities.