The mean and standard deviation of a binomial distribution are 12 and 2 respectively. What is the number of trials?a)2b)12c)18d)24Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Mean and standard deviation are important measures of a binomial distribution. The mean of a binomial distribution is given by the product of the number of trials (n) and the probability of success (p). The standard deviation of a binomial distribution is given by the square root of the product of the number of trials, the probability of success, and the probability of failure (q).

Given that the mean is 12 and the standard deviation is 2, we can set up the following equations:

Mean = n * p
Standard Deviation = sqrt(n * p * q)

We are looking for the number of trials (n). Let's solve the equations to find the value of n.

1. Finding the probability of success (p):
Since the mean is 12, we can write:
12 = n * p
Let's rearrange the equation to solve for p:
p = 12/n

2. Finding the probability of failure (q):
The probability of failure is equal to 1 minus the probability of success:
q = 1 - p
Substituting the value of p:
q = 1 - (12/n)

3. Calculating the standard deviation:
Since the standard deviation is 2, we can write:
2 = sqrt(n * p * q)
Squaring both sides of the equation:
4 = n * p * q

4. Substituting the values of p and q:
4 = n * (12/n) * (1 - (12/n))
Simplifying the equation:
4 = 12 - (144/n) + 48/n
Multiplying through by n:
4n = 12n - 144 + 48
Combining like terms:
8n = 48 + 144
8n = 192
Dividing both sides by 8:
n = 24

Therefore, the number of trials is 24.
Hence, the correct answer is option 'D'

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