There Are 6 Radar Stations And The Probability Of A Single Radar Station Detecting An Enemy Plane Is 0.60 And The Probability Of Not Detecting An Enemy Plane Is 0.40. What Is The Probability That The Number Of Stations That Detect A Plane Is No More

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Introduction

In this problem, we are given 6 radar stations, each with a probability of detecting an enemy plane of 0.60 and a probability of not detecting an enemy plane of 0.40. We want to find the probability that the number of stations that detect a plane is no more than 3. This is a classic problem of binomial probability, where we have a fixed number of independent trials (6 radar stations), each with a constant probability of success (detecting an enemy plane) and failure (not detecting an enemy plane).

Understanding the Problem

To solve this problem, we need to understand the concept of binomial probability. The binomial probability formula is given by:

P(X = k) = (nCk) * (p^k) * (q^(n-k))

where:

  • P(X = k) is the probability of k successes
  • n is the number of trials (6 radar stations)
  • k is the number of successes (detecting an enemy plane)
  • nCk is the number of combinations of n items taken k at a time
  • p is the probability of success (detecting an enemy plane) = 0.60
  • q is the probability of failure (not detecting an enemy plane) = 0.40

Calculating the Probability

We want to find the probability that the number of stations that detect a plane is no more than 3. This means we need to find the probability of 0, 1, 2, or 3 stations detecting a plane.

Probability of 0 Stations Detecting a Plane

The probability of 0 stations detecting a plane is given by:

P(X = 0) = (6C0) * (0.60^0) * (0.40^6) = 1 * 1 * 0.000064 = 0.000064

Probability of 1 Station Detecting a Plane

The probability of 1 station detecting a plane is given by:

P(X = 1) = (6C1) * (0.60^1) * (0.40^5) = 6 * 0.60 * 0.00032 = 0.001536

Probability of 2 Stations Detecting a Plane

The probability of 2 stations detecting a plane is given by:

P(X = 2) = (6C2) * (0.60^2) * (0.40^4) = 15 * 0.36 * 0.01024 = 0.0055296

Probability of 3 Stations Detecting a Plane

The probability of 3 stations detecting a plane is given by:

P(X = 3) = (6C3) * (0.60^3) * (0.40^3) = 20 * 0.216 * 0.064 = 0.027648

Finding the Probability of No More Than 3 Stations Detecting a Plane

To find the probability of no more than 3 stations detecting a plane, we need to add the probabilities of 0, 1, 2, and 3 stations detecting a plane:

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.000064 + 0.001536 + 0.0055296 + 0.027648 = 0.0350776

Conclusion

In this problem, we used the binomial probability formula to find the probability that the number of stations that detect a plane is no more than 3. We calculated the probabilities of 0, 1, 2, and 3 stations detecting a plane and added them to find the final probability. The probability of no more than 3 stations detecting a plane is approximately 0.0350776.

References

  • Binomial probability formula: P(X = k) = (nCk) * (p^k) * (q^(n-k))
  • Combinations formula: nCk = n! / (k! * (n-k)!)

Further Reading

  • Binomial probability: A comprehensive guide to binomial probability, including formulas, examples, and applications.
  • Probability theory: A detailed introduction to probability theory, including concepts, formulas, and examples.
  • Statistics: A comprehensive guide to statistics, including concepts, formulas, and examples.

Introduction

In our previous article, we explored the concept of binomial probability and its application to a problem involving 6 radar stations. We calculated the probability that the number of stations that detect a plane is no more than 3. In this article, we will answer some frequently asked questions related to binomial probability and radar stations.

Q&A

Q: What is binomial probability?

A: Binomial probability is a type of probability that deals with the number of successes in a fixed number of independent trials, where each trial has a constant probability of success and failure.

Q: What is the formula for binomial probability?

A: The formula for binomial probability is:

P(X = k) = (nCk) * (p^k) * (q^(n-k))

where:

  • P(X = k) is the probability of k successes
  • n is the number of trials
  • k is the number of successes
  • nCk is the number of combinations of n items taken k at a time
  • p is the probability of success
  • q is the probability of failure

Q: What is the significance of the number of combinations (nCk) in binomial probability?

A: The number of combinations (nCk) represents the number of ways to choose k items from a set of n items. In binomial probability, it represents the number of ways to choose k successes from a set of n trials.

Q: How do you calculate the number of combinations (nCk)?

A: The number of combinations (nCk) can be calculated using the formula:

nCk = n! / (k! * (n-k)!)

where:

  • n! is the factorial of n
  • k! is the factorial of k
  • (n-k)! is the factorial of (n-k)

Q: What is the difference between binomial probability and other types of probability?

A: Binomial probability is a type of probability that deals with the number of successes in a fixed number of independent trials. Other types of probability, such as geometric probability and hypergeometric probability, deal with different types of random variables and scenarios.

Q: Can you give an example of how to use binomial probability in real-life scenarios?

A: Yes, binomial probability can be used in a variety of real-life scenarios, such as:

  • Quality control: A manufacturer wants to know the probability that a certain number of defective products will be produced in a batch of 100 products.
  • Medical research: A researcher wants to know the probability that a certain number of patients will respond to a new treatment in a clinical trial.
  • Finance: An investor wants to know the probability that a certain number of stocks will increase in value over a certain period of time.

Q: How do you calculate the probability of no more than k successes in a binomial distribution?

A: To calculate the probability of no more than k successes in a binomial distribution, you need to add the probabilities of 0, 1, 2, ..., k successes.

Q: Can you give an example of how to calculate the probability of no more than k successes in a binomial distribution?

A: Yes, let's say we have a binomial distribution with n = 6, p = 0.60, and q = 0.40. We want to calculate the probability of no more than 3 successes. We can calculate the probabilities of 0, 1, 2, and 3 successes and add them together:

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.000064 + 0.001536 + 0.0055296 + 0.027648 = 0.0350776

Conclusion

In this article, we answered some frequently asked questions related to binomial probability and radar stations. We covered topics such as the formula for binomial probability, the significance of the number of combinations, and how to calculate the probability of no more than k successes in a binomial distribution. We also provided examples of how to use binomial probability in real-life scenarios.

References

  • Binomial probability formula: P(X = k) = (nCk) * (p^k) * (q^(n-k))
  • Combinations formula: nCk = n! / (k! * (n-k)!)
  • Binomial distribution: A comprehensive guide to the binomial distribution, including formulas, examples, and applications.

Further Reading

  • Binomial probability: A comprehensive guide to binomial probability, including formulas, examples, and applications.
  • Probability theory: A detailed introduction to probability theory, including concepts, formulas, and examples.
  • Statistics: A comprehensive guide to statistics, including concepts, formulas, and examples.