From three couples, two persons are selected at random.find probability that (i) selected persons are not husband and wife 

Let's solve this problem step by step.

### Step 1: Understand the scenario
You have three couples, meaning there are 6 individuals. You need to select two persons at random, and you want to find the probability that the two selected persons are not husband and wife.

### Step 2: Total number of ways to select two persons
The total number of ways to select 2 persons from 6 individuals can be calculated using the combination formula:
\[
\text{Total number of ways} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15
\]

### Step 3: Number of favorable outcomes (Not a husband and wife pair)
Since there are 3 couples, there are 3 possible husband-wife pairs. If you select a husband-wife pair, that's 1 way for each couple. Therefore, the number of ways to select a husband-wife pair is 3.

The number of ways to select two persons who are **not** a husband-wife pair is given by subtracting the unfavorable outcomes (husband-wife pairs) from the total outcomes:
\[
\text{Number of favorable outcomes} = 15 - 3 = 12
\]

### Step 4: Calculate the probability
The probability that the selected persons are not husband and wife is the ratio of favorable outcomes to the total number of outcomes:
\[
\text{Probability} = \frac{12}{15} = \frac{4}{5}
\]

### Final Answer:
The probability that the selected persons are not husband and wife is \(\frac{4}{5}\).

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Here are some related questions:
1. What is the probability that the selected persons are husband and wife?
2. How does the probability change if there are 4 couples instead of 3?
3. What if you select 3 persons at random? What is the probability that none of them are husband and wife?
4. How would the probability change if the selection was done without replacement?
5. What if you selected two persons such that one is from the first couple and the other from the second couple?

**Tip:** When dealing with probabilities involving combinations, always consider how the total number of outcomes and the number of favorable outcomes relate to each other to ensure accuracy.

From three couples, two persons are selected at random. Find the probability that (i) selected persons are not husband and wife.

Explore the probability problem of selecting two persons at random from three couples where they are not husband and wife. Learn the key concepts of combinatorics and probability with a step-by-step solution. Ideal for students in Grades 9-12.

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