Understanding Binomial Probability: Calculating the Probability of Hitting a Target at Least Twice

Probabilities play a critical role in various scenarios ranging from simple games and events to complex real-world situations. One such application is calculating the probability of hitting a target a certain number of times in a series of attempts. In this article, we will delve into the calculation of the probability of hitting a target at least twice in four attempts, using the binomial probability distribution formula.

Binomial Probability Distribution Formula

The binomial probability formula helps us determine the probability of achieving a specific number of successes in a given number of trials. The formula is given by:

P(X k) C(n, k) * p^k * q^(n-k)

where:

C(n, k) is the binomial coefficient, which can be calculated as C(n, k) n! / [k! * (n - k)!] n is the number of trials k is the number of successes p is the probability of success q is the probability of failure, given by q 1 - p

Calculating the Probability of Hitting a Target at Least Twice in Four Attempts

Given:

The probability of hitting the target success is p 1/4 The probability of missing the target failure is q 3/4 The number of attempts trials is n 4 We need to find the probability of hitting the target at least twice (i.e., 2, 3, or 4 times)

Step 1: Calculate the probabilities for each case using the binomial probability distribution formula.

Probability of Hitting the Target Exactly Twice (k 2)

P(X 2) C(4, 2) * (1/4)^2 * (3/4)^2

C(4, 2) 4! / [2! * (4 - 2)!] 6

P(X 2) 6 * (1/16) * (9/16) 6 * (9/256) 54/256 27/128

Probability of Hitting the Target Exactly Three Times (k 3)

P(X 3) C(4, 3) * (1/4)^3 * (3/4)^1

C(4, 3) 4! / [3! * (4 - 3)!] 4

P(X 3) 4 * (1/64) * (3/4) 4 * (3/256) 12/256 3/64

Probability of Hitting the Target Four Times (k 4)

P(X 4) C(4, 4) * (1/4)^4 * (3/4)^0

C(4, 4) 4! / [4! * (4 - 4)!] 1

P(X 4) 1 * (1/256) * 1 1/256

Summing Up the Probabilities

To find the probability of hitting the target at least twice, we add the probabilities of hitting the target twice, three times, and four times:

P(X ge; 2) P(X 2) P(X 3) P(X 4)

P(X ge; 2) 54/256 12/256 1/256 67/256

Therefore, the probability that the man hits the target at least twice in four attempts is:

Final Result: 67/256

Example Calculation with Different Parameters

Now, consider a similar scenario where n 7 and P 1/4. Using the same formula:

P(X 2) C(7, 2) * (1/4)^2 * (3/4)^5

C(7, 2) 7! / [2! * (7 - 2)!] 21

P(X 2) 21 * (1/16) * (243/1024) 21 * (243/16384) 5103/16384

This example illustrates how the binomial probability formula can be applied to real-world scenarios involving multiple attempts and different probabilities.

Conclusion

Understanding the binomial probability distribution is crucial for calculating outcomes in scenarios with repeated trials and fixed probabilities. By following the steps outlined in this article, you can effectively determine the probability of hitting a target at least twice in four attempts or apply similar calculations with different parameters.