The odds in favour of a person hitting a target are 3 to 5.\nThe odds against another person hitting the same target are 3 to 2.\nEach of them fires once at the target independently.\nWhat is the probability that both of them hit the target?

Introduction / Context:
This question checks your understanding of odds, conversion of odds into probabilities, and the multiplication rule for independent events. Two people shoot at the same target under different odds conditions, and you must find the probability that both shots are successful.

Given Data / Assumptions:

• Odds in favour of the first shooter hitting the target are 3 to 5.
• Odds against the second shooter hitting the target are 3 to 2.
• Each shooter fires exactly once.
• The events of their hits are independent.
• We want the probability that both shooters hit the target.

Concept / Approach:
For odds in favour a : b, the probability of success is a / (a + b). For odds against a : b, the probability of success is b / (a + b), because a corresponds to failure and b corresponds to success. Once we find the individual probabilities of hitting, we use the multiplication rule: P(both hit) = P(first hits) * P(second hits)

Step-by-Step Solution:
For the first shooter, odds in favour are 3 : 5. Probability first shooter hits = 3 / (3 + 5) = 3 / 8. For the second shooter, odds against are 3 : 2. So probability second shooter hits = 2 / (3 + 2) = 2 / 5. Now, P(both hit) = (3 / 8) * (2 / 5) = 6 / 40. Simplify 6 / 40 = 3 / 20. Thus, the probability that both shooters hit the target is 3/20.

Verification / Alternative check:
We can verify by converting to decimals. Probability for the first shooter is 0.375, for the second shooter is 0.4. Multiplying gives 0.375 * 0.4 = 0.15. Converting 3/20 to decimal gives 0.15, which is consistent. This confirms that our fraction is correct.

Why Other Options Are Wrong:
Option 1/5 equals 0.2, which is larger than the correct probability of 0.15. Option 1/20 is 0.05 and would require either much smaller odds or dependent events. Option 3/40 is 0.075 and corresponds to halving the correct probability accidentally.

Common Pitfalls:
Typical mistakes include misinterpreting odds in favour or against, reversing success and failure, or adding probabilities instead of multiplying for independent events. Another frequent error is simplifying odds directly as probabilities without dividing by the total of both parts. Always convert odds to probabilities carefully before applying multiplication rules.

Final Answer:
The probability that both people hit the target is 3/20.