Binomial Distribution

Step 1: Define Probabilities of Detection and Failure
Let be the probability of detecting an enemy plane in a single scan. Given that detection occurs once in 10 scans, we have: The probability of failing to detect the plane in a single scan is: Step 2: Define the Probability of Not Detecting at Least Twice in Four Scans
The possible cases where detection occurs fewer than two times in four scans are:
1. No detection in all four scans.
2. Exactly one detection in four scans.
Case 1: No Detection in Four Scans
The probability of missing detection in all four scans: Case 2: Exactly One Detection in Four Scans
The probability of detecting exactly once in four scans follows the binomial distribution: Step 3: Compute the Final Probability
The required probability is the sum of these two cases: Thus, the correct answer is .

Step 1: Define the Binomial Distribution A binomial distribution is given by: where: - is the number of trials, - is the probability of success. The mean and standard deviation of a binomial distribution are:

Step 2: Given Conditions and Forming Equations We are given: Substituting and : Squaring both sides:
Step 3: Solve for and Let , so we rewrite the equations: Squaring both sides: Rewriting from the first equation: Substituting in the second equation: Expanding: Since , substituting back into the first equation: Squaring: Since , we find :
Step 4: Compute Probability Ratio The binomial probability formula is: For : For : Taking their ratio: Thus:

To solve the problem, we use the properties of the Binomial distribution :
- Mean ( ): ,
- Variance ( ): , where . Given: 1. The sum of the mean and variance is 5: 2. The product of the mean and variance is 6: Step 1: Simplify the equations From the first equation: From the second equation: Step 2: Express in terms of Since , substitute into the first equation: Step 3: Solve for and From the second equation: Substitute : From the first equation, solve for : Substitute into the second equation: Simplify: Multiply through by : Expand : Simplify: Rearrange: Step 4: Solve the cubic equation The cubic equation can be solved numerically or by inspection. One solution is , but this is not valid since must be between 0 and 1. Another solution is . Substitute into : Thus: Step 5: Compute Substitute , , and : Find a common denominator: Multiply by 6: This does not match any of the options. Let's re
-examine the problem. Step 6: Re
-examining the problem The correct approach is to use the given equations: 1. , 2. . Substitute : Let . Then: From the first equation: Substitute into the second equation: Simplify: Multiply through by : Expand : Simplify: Rearrange: Solve the quadratic equation: Thus: Since must be between 0 and 1, . Substitute into : Thus: Now, compute : Multiply by 6: Final Answer: