Binomial Distribution Practice Questions

The binomial distribution is a probability distribution that gives the probability of obtaining a certain number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.

Example 1: A die is thrown 6 times and if getting an even number is a success, what is the probability of getting
(i) 4 Successes
(ii) No success

Solution:

Given: n = 6, p = 3/6 = 1/2, and q = 1 - 1/2 = 1/2

P(X = r) = ⁿC_rp^rq^n-r

(i) P(X = 4) = ⁶C₄(1/2)⁴(1/2)² = 15/64
(ii) P(X = 0) = ⁶C₀(1/2)⁰(1/2)⁶ = 1/64

Example 2: A coin is tossed 4 times what is the probability of getting at least 2 heads?

Solution:

Given: n = 4
Probability of getting head in each trial, p = 1/2 ⇒ q = 1 - 1/2 = 1/2

P(X = r) = ⁴C_r(1/2)^r(1/2)^4-r
⇒ P(X = r) = ⁴C_r(1/2)⁴ {Using the laws of Exaponents}

And we know, Probability of getting at least 2 heads = P(X ≥ 2)

⇒ Probability of getting at least 2 heads = P(X = 2) + P(X = 3) + P(X = 4)
⇒ Probability of getting at least 2 heads = ⁴C₂(1/2)⁴ + ⁴C₃(1/2)⁴ + ⁴C₄(1/2)⁴
⇒ Probability of getting at least 2 heads = (⁴C₂ + ⁴C₃ + ⁴C₄)(1/2)⁴
⇒ Probability of getting at least 2 heads = 11(1/2)⁴ = 11/16

Example 3: A pair ofdice is thrown 6 times and getting sum 5 is a success then what is the probability of getting (i) no success (ii) two success (iii) at most two success

Solution:

Given: n = 6

5 can be obtained in 4 ways (1, 4) (4, 1) (2, 3) (3, 2)

Probability of getting the sum 5 in each trial, p = 4/36 = 1/9
Probability of not getting sum 5 = 1 - 1/9 = 8/9

(i) Probability of getting no success, P(X = 0) = ⁶C₀(1/9)⁰(8/9)⁶ = (8/9)⁶
(ii) Probability of getting two success, P(X = 2) = ⁶C₂(1/9)²(8/9)⁴ = 15(8⁴/9⁶)
(iii) Probability of getting at most two successes, P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

⇒ P(X ≤ 2) = (8/9)⁶ + 6(8⁵/9⁶) + 15(8⁴/9⁶)

Practice Problems

1. A box has 5 red, 7 black,? and 8 white balls. If three balls are drawn one by one with replacement what is the probability that all,

i) all are white
ii) all are red
iii) all are black

2. What is the probability distribution of the number of tails when three coins are tossed together?

3. A die is thrown three times what is the probability distribution of getting six?

4. A coin is tossed 4 times then what is the probability distribution of getting head.