A bag contains 3 red balls, 5 yellow balls and 7 pink balls. If one ball is drawn at random from the bag, what is the probability that the ball drawn is either pink or red?

Introduction / Context:
This is a simple one step probability problem involving a single draw from a bag containing coloured balls. The event of interest is that the ball drawn is either pink or red. We are given the exact counts of each colour, and the drawing is at random, so each ball has the same chance of being chosen. Such questions test basic understanding of favourable outcomes divided by total outcomes.

Given Data / Assumptions:

Red balls: 3.

Yellow balls: 5.

Pink balls: 7.

Total balls = 3 + 5 + 7 = 15.

One ball is drawn at random.

We need the probability that the ball is pink or red.

Concept / Approach:
The probability of an event is defined as the number of favourable outcomes divided by the total number of equally likely outcomes. Here, the outcomes are the specific balls in the bag. A favourable outcome occurs when the chosen ball is either red or pink. Because these colour groups do not overlap and there is no restriction about yellow, we simply add the counts of red and pink balls to find the number of favourable outcomes.

Step-by-Step Solution:
Step 1: Compute total number of balls = 3 red + 5 yellow + 7 pink = 15.Step 2: Count favourable balls: these are balls that are either red or pink.Step 3: Number of red balls = 3.Step 4: Number of pink balls = 7.Step 5: Total favourable balls = 3 + 7 = 10.Step 6: Probability = favourable / total = 10 / 15.Step 7: Simplify 10 / 15 by dividing numerator and denominator by 5 to get 2 / 3.

Verification / Alternative check:
We can verify by computing complementary probabilities. The only colour that does not count as favourable is yellow. There are 5 yellow balls, so the probability of drawing yellow is 5 / 15 = 1 / 3. The probability of drawing red or pink is therefore 1 - 1 / 3 = 2 / 3, which matches the direct calculation and confirms the correctness of the answer.

Why Other Options Are Wrong:
The fraction 3/5 would correspond to 9 favourable balls out of 15, which is not correct because there are 10. The value 3/8 does not relate to the actual counts and would imply a total of 8 outcomes. The probability 1/8 is far too small and ignores the fact that most balls are either pink or red. The fraction 7/15 would be correct if only pink balls were favourable, but here both pink and red count, so we must include both colours.

Common Pitfalls:
A common mistake is to miscount total balls or to forget one of the colours when computing favourable outcomes. Some learners overcomplicate the problem by trying to use combinations, even though this is a single simple selection. Always remember that when each individual object is equally likely, you can often work directly with counts rather than more complex formulas.

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