# Probability and Statistics Questions and Answers – Binomial Distribution

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This set of Probability and Statistics Multiple Choice Questions & Answers (MCQs) focuses on “Binomial Distribution”.

1. In a Binomial Distribution, if ‘n’ is the number of trials and ‘p’ is the probability of success, then the mean value is given by ___________
a) np
b) n
c) p
d) np(1-p)

Explanation: For a discrete probability function, the mean value or the expected value is given by
Mean (μ)=$$\sum_{x=0}^{n}xp(x)$$
For Binomial Distribution P(x)=nCx px q(n-x), substitute in above equation and solve to get
µ = np.

2. In a Binomial Distribution, if p, q and n are probability of success, failure and number of trials respectively then variance is given by ___________
a) np
b) npq
c) np2q
d) npq2

Explanation: For a discrete probability function, the variance is given by
Variance (V)=$$\sum_{x=0}^n x^2p(x)-\mu^2$$
Where µ is the mean, substitute P(x)=nCx px q(n-x) in the above equation and put µ = np to obtain
V = npq.

3. If ‘X’ is a random variable, taking values ‘x’, probability of success and failure being ‘p’ and ‘q’ respectively and ‘n’ trials being conducted, then what is the probability that ‘X’ takes values ‘x’? Use Binomial Distribution
a) P(X = x) = nCx px qx
b) P(X = x) = nCx px q(n-x)
c) P(X = x) = xCn qx p(n-x)
d) P(x = x) = xCn pn qx

Explanation: It is the formula for Binomial Distribution that is asked here which is given by P(X = x) = nCx px q(n-x).

4. If ‘p’, ‘q’ and ‘n’ are probability pf success, failure and number of trials respectively in a Binomial Distribution, what is its Standard Deviation?
a) $$\sqrt{np}$$
b) $$\sqrt{pq}$$
c) (np)2
d) $$\sqrt{npq}$$

Explanation: The variance (V) for a Binomial Distribution is given by V = npq
Standard Deviation = $$\sqrt{variance} = \sqrt{npq}$$.

5. In a Binomial Distribution, the mean and variance are equal.
a) True
b) False

Explanation: Mean = np
Variance = npq
∴ Mean and Variance are not equal.

6. It is suitable to use Binomial Distribution only for ___________
a) Large values of ‘n’
b) Fractional values of ‘n’
c) Small values of ‘n’
d) Any value of ‘n’

Explanation: As the value of ‘n’ increases, it becomes difficult and tedious to calculate the value of nCx.

7. For larger values of ‘n’, Binomial Distribution ___________
a) loses its discreteness
b) tends to Poisson Distribution
c) stays as it is
d) gives oscillatory values

Explanation: P(x)=$$\lim_{n\rightarrow\infty}c_x^n p^x q^{n-x}=\frac{e^{-m}m^x}{x!}$$
Where m = np is the mean of Poisson Distribution.

8. In a Binomial Distribution, if p = q, then P(X = x) is given by?
a) nCx (0.5)n
b) nCn (0.5)n
c) nCx p(n-x)
d) nCn p(n-x)

Explanation: If p = q, then p = 0.5
Substituting in P(x)=nCx px q(n-x) we get nCn (0.5)n.

9. Binomial Distribution is a ___________
a) Continuous distribution
b) Discrete distribution
c) Irregular distribution
d) Not a Probability distribution