Question

A bag contains some red and some white balls. A ball is drawn at random from the bag. If the probability of getting a red ball is \(\frac{2}{7}\), then the probability of getting a white ball is

• A. \(\frac{1}{14}\)
• B. \(\frac{5}{7}\)
• C. \(\frac{1}{7}\)
• D. \(\frac{2}{7}\)

Hint:

For any two complementary outcomes, if the probability of one is \(\frac{x}{y}\), the probability of the other is always \(\frac{y - x}{y}\).
Here, \(\frac{7 - 2}{7} = \frac{5}{7}\), which can be determined mentally in a split second!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is based on Probability, specifically dealing with complementary events.
The bag contains only two types of balls: red and white.
When a ball is drawn at random, the event of drawing a red ball and the event of drawing a white ball are mutually exclusive and collectively exhaustive.
We are given the probability of drawing a red ball and need to find the probability of drawing a white ball.

Step 2: Key Formula or Approach:
Since the bag contains only red and white balls, the sum of their probabilities must equal 1: \[ P(\text{Red}) + P(\text{White}) = 1 \] Therefore, the probability of drawing a white ball is the complement of drawing a red ball: \[ P(\text{White}) = 1 - P(\text{Red}) \]

Step 3: Detailed Explanation:
1. Write down the given probability of drawing a red ball: \[ P(\text{Red}) = \frac{2}{7} \] 2. Since there are only red and white balls in the bag, the occurrence of one means the non-occurrence of the other.
3. Apply the complementary probability formula: \[ P(\text{White}) = 1 - P(\text{Red}) \] 4. Substitute the value of \(P(\text{Red})\) into the formula: \[ P(\text{White}) = 1 - \frac{2}{7} \] 5. Take the common denominator to subtract the fraction: \[ P(\text{White}) = \frac{7 - 2}{7} = \frac{5}{7} \]

Step 4: Final Answer:
The probability of getting a white ball is \(\frac{5}{7}\).
Therefore, the correct option is (B).