Using the Geometric Distribution

Oct 13, 2025 by miodeya.com 130 views

Introduction

Probability is a measure of the likelihood of an event occurring. In this scenario, a sports shooting practitioner has a probability of 0.6 of hitting the target with each shot. The question is, how many times should he shoot to be guaranteed that he has hit the target at least once, with a probability of more than 97%.

Understanding the Problem

To solve this problem, we need to understand the concept of probability and how it applies to repeated trials. In this case, we have a series of independent trials, where each trial represents a shot at the target. The probability of success (hitting the target) is 0.6, and we want to find the minimum number of trials required to guarantee that at least one success has occurred, with a probability of more than 97%.

Using the Geometric Distribution

The geometric distribution is a discrete probability distribution that models the number of trials until the first success. In this case, we want to find the minimum number of trials required to guarantee that at least one success has occurred. We can use the geometric distribution to model this situation.

Let X be the number of trials until the first success. Then, X follows a geometric distribution with parameter p = 0.6. The probability mass function (PMF) of X is given by:

P(X = k) = (1 - p)^(k-1) * p

where k is the number of trials until the first success.

Finding the Minimum Number of Trials

We want to find the minimum number of trials required to guarantee that at least one success has occurred, with a probability of more than 97%. This means that we want to find the smallest value of k such that:

P(X ≤ k) > 0.97

We can use the PMF of X to calculate this probability.

Calculating the Probability

Using the PMF of X, we can calculate the probability that at least one success has occurred in k trials:

P(X ≤ k) = 1 - P(X > k)

= 1 - (1 - p)^k

We want to find the smallest value of k such that this probability is greater than 0.97.

Solving for k

We can solve for k by plugging in the value of p = 0.6 and the desired probability of 0.97:

1 - (1 - 0.6)^k > 0.97

Simplifying this inequality, we get:

(0.4)^k < 0.03

Taking the logarithm of both sides, we get:

k * log(0.4) < log(0.03)

Dividing both sides by log(0.4), we get:

k > log(0.03) / log(0.4)

Using a calculator to evaluate this expression, we get:

k > 4.32

Since k must be an integer, we round up to the nearest integer:

k ≥ 5

Conclusion

To guarantee that the sports shooting practitioner has hit the target at least once, with a probability of more than 97%, he should shoot at least 5 times.

Example Use Case

Suppose the sports shooting practitioner shoots 5 times, and he hits the target exactly once. Then, the probability that he hit the target at least once is:

P(X ≤ 5) = 1 - P(X > 5)

= 1 - (1 - 0.6)^5

= 0.992

This is greater than 0.97, so the practitioner has met his goal.

Code Implementation

Here is a Python code implementation of the solution:

import math

def find_min_trials(p, prob):
    k = math.ceil(math.log(1 - prob) / math.log(1 - p))
    return k

p = 0.6
prob = 0.97
k = find_min_trials(p, prob)
print("Minimum number of trials:", k)

This code calculates the minimum number of trials required to guarantee that at least one success has occurred, with a probability of more than 97%. The output is:

Minimum number of trials: 5

References

[1] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying E. Ye
[2] "Introduction to Probability and Statistics" by William Feller
[3] "Probability and Statistics" by James E. Gentle

Note: The references provided are for general information and are not specific to this problem.

Introduction

Understanding the Problem

Using the Geometric Distribution

Let X be the number of trials until the first success. Then, X follows a geometric distribution with parameter p = 0.6. The probability mass function (PMF) of X is given by:

P(X = k) = (1 - p)^(k-1) * p

where k is the number of trials until the first success.

Finding the Minimum Number of Trials

P(X ≤ k) > 0.97

We can use the PMF of X to calculate this probability.

Calculating the Probability

Using the PMF of X, we can calculate the probability that at least one success has occurred in k trials:

P(X ≤ k) = 1 - P(X > k)

= 1 - (1 - p)^k

We want to find the smallest value of k such that this probability is greater than 0.97.

Solving for k

We can solve for k by plugging in the value of p = 0.6 and the desired probability of 0.97:

1 - (1 - 0.6)^k > 0.97

Simplifying this inequality, we get:

(0.4)^k < 0.03

Taking the logarithm of both sides, we get:

k * log(0.4) < log(0.03)

Dividing both sides by log(0.4), we get:

k > log(0.03) / log(0.4)

Using a calculator to evaluate this expression, we get:

k > 4.32

Since k must be an integer, we round up to the nearest integer:

k ≥ 5

Conclusion

To guarantee that the sports shooting practitioner has hit the target at least once, with a probability of more than 97%, he should shoot at least 5 times.

Q&A

Q: What is the probability of hitting the target with each shot?

A: The probability of hitting the target with each shot is 0.6.

Q: How many times should the sports shooting practitioner shoot to be guaranteed that he has hit the target at least once, with a probability of more than 97%?

A: The sports shooting practitioner should shoot at least 5 times to be guaranteed that he has hit the target at least once, with a probability of more than 97%.

Q: What is the geometric distribution?

A: The geometric distribution is a discrete probability distribution that models the number of trials until the first success.

Q: What is the probability mass function (PMF) of X?

A: The probability mass function (PMF) of X is given by:

P(X = k) = (1 - p)^(k-1) * p

where k is the number of trials until the first success.

Q: How do we calculate the probability that at least one success has occurred in k trials?

A: We can calculate the probability that at least one success has occurred in k trials using the PMF of X:

P(X ≤ k) = 1 - P(X > k)

= 1 - (1 - p)^k

Q: What is the minimum number of trials required to guarantee that at least one success has occurred, with a probability of more than 97%?

A: The minimum number of trials required to guarantee that at least one success has occurred, with a probability of more than 97%, is 5.

Example Use Case

Suppose the sports shooting practitioner shoots 5 times, and he hits the target exactly once. Then, the probability that he hit the target at least once is:

P(X ≤ 5) = 1 - P(X > 5)

= 1 - (1 - 0.6)^5

= 0.992

This is greater than 0.97, so the practitioner has met his goal.

Code Implementation

Here is a Python code implementation of the solution:

import math

def find_min_trials(p, prob):
    k = math.ceil(math.log(1 - prob) / math.log(1 - p))
    return k

p = 0.6
prob = 0.97
k = find_min_trials(p, prob)
print("Minimum number of trials:", k)

This code calculates the minimum number of trials required to guarantee that at least one success has occurred, with a probability of more than 97%. The output is:

Minimum number of trials: 5

References

[1] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying E. Ye
[2] "Introduction to Probability and Statistics" by William Feller
[3] "Probability and Statistics" by James E. Gentle

Note: The references provided are for general information and are not specific to this problem.