Binomial Distribution in Probability – Solved Examples
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) = nCrprqn-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) = 4Cr(1/2)r(1/2)4-r
⇒ P(X = r) = 4Cr(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 of die 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)
Binomial Distribution in Probability
Binomial Distribution in Probability gives information about two types of possible outcomes i.e. Success or Failure. Binomial Probability Distribution is a discrete probability distribution used for the events that give results in ‘Yes or No’ or ‘Success or Failure’.
Binomial distribution is calculated by multiplying the probability of success raised to the power of the number of successes and the probability of failure raised to the power of the difference between the number of successes and the number of trials.
In this article, we will learn about binomial probability distributions, binomial distribution formulas, binomial distribution meaning, and properties of a binomial distribution.
Table of Content
- What is Binomial Distribution in Probability?
- Binomial Distribution Definition
- Binomial Distribution Meaning
- Binomial Distribution Formula
- Binomial Distribution Calculation
- Binomial Distribution Examples
- Bernoulli Trials in Binomial Distribution
- Binomial Random Variable
- Binomial Distribution Table
- Binomial Distribution Graph
- Binomial Distribution in Statistics
- Measure of Central Tendency for Binomial Distribution
- Binomial Distribution Mean
- Binomial Distribution Variance
- Binomial Distribution Standard Deviation
- Binomial Distribution Properties
- Binomial Distribution Applications
- Negative Binomial Distribution
- Negative Binomial Distribution Formula
- Binomial Distribution vs Normal Distribution
- Binomial Distribution in Probability – Solved Examples
- Practice Problems on Binomial Distribution in Probability
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