A bag contains 15 red balls and 20 black balls. Each ball is numbered either 1 or 2 or 3. 20% of the red balls are numbered 1 and 40% of them are numbered 3. Similarly among the black balls, 45% are numbered 2 and 30% are numbered 3. A boy picks a ball at random. He wins if the ball is red and numbered 3 or if it is black and numbered 1 or 2. What are the chances of his wining?

Correct Option (3)

To determine the probability of winning, we first calculate the total number of balls and the number of balls that satisfy the winning conditions.

• Total number of red balls = 15
• Total number of black balls = 20
• Total number of balls in the bag = 15 + 20 = 35

The winning conditions are:

1. The ball is red and numbered 3.
2. The ball is black and numbered 1 or 2.

Calculating the number of balls for each winning condition:

• Number of red balls numbered 3: 40% of 15 = (40/100) * 15 = 6 balls.
• Number of black balls numbered 1: Among black balls, 45% are numbered 2 and 30% are numbered 3. Therefore, the percentage of black balls numbered 1 is 100% - 45% - 30% = 25%. Number of black balls numbered 1 = 25% of 20 = (25/100) * 20 = 5 balls.
• Number of black balls numbered 2: 45% of 20 = (45/100) * 20 = 9 balls.

Total number of black balls numbered 1 or 2 = 5 (numbered 1) + 9 (numbered 2) = 14 balls.

Total number of favorable outcomes (winning balls) = (Red balls numbered 3) + (Black balls numbered 1 or 2) = 6 + 14 = 20 balls.

The probability of winning is calculated as the ratio of favorable outcomes to the total number of balls:

Probability of winning = (Favorable outcomes) / (Total balls) = 20 / 35 = 4 / 7.

Incorrect Options:

The other options represent numerical values that do not align with the calculated probability of 4/7. These options are arithmetically incorrect based on the given conditions and the principles of probability calculation.