Question

If the mean and standard deviation of a binomial distribution are 12 and 2 respectively, then the value of its parameter p is
A. $$\dfrac{1}{2}$$
B. $$\dfrac{1}{3}$$
C. $$\dfrac{2}{3}$$
D. $$\dfrac{1}{4}$$

Answer 1

Hint: In this question it is given that the mean and standard deviation of a binomial distribution are 12 and 2 respectively, we have to find the value of the parameter p. So to find the solution we need to know that the mean and standard deviation of a binomial distribution is $$np$$ and $$\sqrt{np\left( 1-p\right) }$$ respectively, where n is the number of trials in a binomial experiment and p is the probability of success on an individual trial. So by using these we will get our required solution.

Complete step-by-step solution:
given that,
Mean(m) = 12 and standard deviation(sd) = 2
Als as we know that for any binomial distribution the mean and standard deviations are,
m = np and sd = $$\sqrt{np\left( 1-p\right) }$$
Therefore, we can write,
 np = 12………….........(1) and
 $$\sqrt{np\left( 1-p\right) } =2$$
 $$np\left( 1-p\right) =2^{2}$$
 $$np\left( 1-p\right) =4$$………………..(2)
Now putting the value of ‘np’ in equation (2) we get,
$$np\left( 1-p\right) =4$$
$$\Rightarrow 12\left( 1-p\right) =4$$
$$\Rightarrow \left( 1-p\right) =\dfrac{4}{12}$$ [dividing both side by 12]
$$\Rightarrow \left( 1-p\right) =\dfrac{1}{3}$$
$$\Rightarrow -1+p=-\dfrac{1}{3}$$ [multiplying ‘-1’ in the both side of the equation]
$$\Rightarrow p=-\dfrac{1}{3} +1$$
$$\Rightarrow p=1-\dfrac{1}{3}$$
$$\Rightarrow p=\dfrac{3-1}{3}$$
$$\Rightarrow p=\dfrac{2}{3}$$
Therefore the value of the parameter p is $$\dfrac{2}{3}$$.
Hence the correct option is option C.

Note: While solving this type of question you need to know that in probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes/no question, and each with its own boolean-valued outcome: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q = 1 − p).
In general, if the random variable X follows the binomial distribution with parameters $$n\in \mathbf{N}$$ and $$p\in \left[ 0,1\right] $$, we write $$\mathrm{X} \sim B\left( n,p\right) $$. The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:
$$\mathrm{P} \left( \mathrm{X} =k\right) =\ ^{n} C_{k}\ p^{k}q^{n-k}$$.