Model Answer
0 min readIntroduction
Probability, a fundamental concept in statistics and mathematics, quantifies the likelihood of an event occurring. It is widely used in various fields, including risk assessment, decision-making, and predictive modeling. In this problem, we are dealing with independent events – the success or failure of each person hitting the target does not influence the others. Calculating the probability of combined events requires understanding the principles of multiplication of probabilities for independent events. This question assesses the candidate’s ability to apply these principles to a practical scenario.
Understanding the Problem
Let A, B, and C represent the events that persons A, B, and C hit the target, respectively. We are given the following probabilities:
- P(A) = 4/5 = 0.8
- P(B) = 3/4 = 0.75
- P(C) = 2/3 ≈ 0.667
We need to calculate the probabilities of three different scenarios.
(A) Probability that A, B, and C all hit the target
Since the events are independent, the probability of all three hitting the target is the product of their individual probabilities:
P(A ∩ B ∩ C) = P(A) * P(B) * P(C) = (4/5) * (3/4) * (2/3) = 24/60 = 2/5 = 0.4
(B) Probability that B and C hit the target, but A does not
First, we need to find the probability that A does *not* hit the target: P(A') = 1 - P(A) = 1 - (4/5) = 1/5 = 0.2
Now, we calculate the probability of B and C hitting the target, and A missing the target:
P(A' ∩ B ∩ C) = P(A') * P(B) * P(C) = (1/5) * (3/4) * (2/3) = 6/60 = 1/10 = 0.1
(C) Probability that C and A hit the target, but B does not
First, we need to find the probability that B does *not* hit the target: P(B') = 1 - P(B) = 1 - (3/4) = 1/4 = 0.25
Now, we calculate the probability of C and A hitting the target, and B missing the target:
P(C ∩ A ∩ B') = P(C) * P(A) * P(B') = (2/3) * (4/5) * (1/4) = 8/60 = 2/15 ≈ 0.133
Summary of Results
| Scenario | Probability |
|---|---|
| A, B, and C all hit | 0.4 |
| B and C hit, A misses | 0.1 |
| C and A hit, B misses | 2/15 (≈ 0.133) |
Conclusion
In conclusion, we have successfully calculated the probabilities for each of the given scenarios using the fundamental principles of probability. The key takeaway is the application of the multiplication rule for independent events. Understanding these concepts is crucial for analyzing and predicting outcomes in various real-world situations, from quality control to risk management. The ability to accurately assess probabilities is a valuable skill in many fields.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.