A bag contains 3 black balls, 4 white balls and 5 red balls.\nOne ball is drawn at random from the bag.\nWhat is the probability that the ball drawn is either black or red?

Introduction / Context:
Here we again consider a simple bag of coloured balls. The task is to find the probability that a randomly drawn ball is either black or red. This is a basic example of adding probabilities for mutually exclusive events.

Given Data / Assumptions:

• Black balls = 3.
• White balls = 4.
• Red balls = 5.
• Total balls = 3 + 4 + 5 = 12.
• One ball is selected at random.
• We require the probability that the ball is black or red.

Concept / Approach:
The events "ball is black" and "ball is red" are mutually exclusive. Therefore, the probability that the ball is black or red is the sum of their individual probabilities. Using the formula: Probability = favourable outcomes / total outcomes. we can count the favourable balls and then divide by the total number of balls.

Step-by-Step Solution:
Total balls = 12. Black balls = 3. Red balls = 5. Favourable balls (black or red) = 3 + 5 = 8. Probability = 8 / 12. Simplify 8 / 12 = 2 / 3. Therefore, the required probability is 2/3.

Verification / Alternative check:
An alternative perspective is to find the probability that the ball is white and subtract from 1. Probability of a white ball is 4 / 12 = 1 / 3. Thus, probability of black or red is 1 - 1 / 3 = 2 / 3. This matches the main method, giving extra confidence in the result.

Why Other Options Are Wrong:
Option 1/4 equals 3 / 12, the probability of drawing a black ball only. Option 5/12 corresponds to just the red balls. Option 1/2 would imply 6 favourable outcomes, which does not match the total of 8 black or red balls.

Common Pitfalls:
Learners often forget to include both colours in the favourable count or mistakenly divide by the number of colours instead of the number of balls. Another issue is not simplifying the fraction fully, which can make it harder to compare with the given options.

Final Answer:
The probability that the ball drawn is black or red is 2/3.