Calculating the Likelihood of an Archer Hitting a Target at a Shooting Range

Archer enthusiasts and casual observers often wonder about the odds of hitting a target at a shooting range. This article explores the mathematical framework behind calculating such probabilities, particularly when weather conditions are a factor. We'll use specific examples and formulas to illustrate the process.

Understanding the Problem

The probability that an archer can hit a target is influenced by the weather conditions. In this scenario, we are given two key pieces of information:

When the weather is clear, the probability of hitting the target is (1/6). The probability that the weather is clear is (1/8).

Applying the Law of Total Probability

To determine the overall probability that the archer hits the target, we need to account for both clear and non-clear weather conditions. This can be done using the law of total probability, which states:

(P(H) P(H|C) cdot P(C) P(H|neg C) cdot P(neg C))

Scenario 1: Clear Weather

Let's denote the event of hitting the target as (H).

When the weather is clear (denoted as (C)):

(P(H|C) 1/6) (P(C) 1/8)

Scenario 2: Non-Clear Weather

In the case of non-clear weather (denoted as (neg C)), we assume the probability of hitting the target is 0, as the weather conditions are a significant factor in archery performance.

(P(H|neg C) 0) (P(neg C) 1 - P(C) 1 - 1/8 7/8)

Calculating the Total Probability

Now, we substitute these values into the law of total probability formula:

(P(H) P(H|C) cdot P(C) P(H|neg C) cdot P(neg C))

(P(H) left(1/6 cdot 1/8right) left(0 cdot 7/8right))

(P(H) 1/48 0 1/48)

Therefore, the probability that the archer hits the target is (1/48).

Bayesian Approach

The Bayesian approach complements the law of total probability. It involves calculating the prior probability of an event and then updating it based on new evidence.

Given the probability of hitting the target in clear weather ((P(H|C) 1/6)) and the probability of clear weather ((P(C) 1/8)), the probability can be calculated as follows:

(P(H) P(H|C) cdot P(C) 1/6 cdot 1/8 1/48)

Thus, the probability of the archer hitting the target is (0.0208).

Conclusion

The problem can be summarized as follows:

Probability of hitting the target (P(H|C) 1/6) Probability of clear weather (P(C) 1/8) Probability of hitting the target given non-clear weather (P(H|neg C) 0)

Using the law of total probability, the overall probability of hitting the target is (1/48), or about (0.0208).

This article has provided a detailed explanation of how to calculate the probability of an archer hitting a target at a shooting range, considering both clear and non-clear weather conditions.