Binomial Distribution Formula

About Binomial Distribution Formula

In statistics, the binomial distribution is a regularly used discrete distribution. In contrast to a binomial distribution, the normal distribution is a continuous distribution. Given a success probability of 'p' for each trial at the experiment, the binomial distribution represents the chance of 'x' successes in 'n' trials.

What Is the Binomial Distribution Formula?

For each random variable X, the binomial distribution formula is:

P(x :n,p) = ⁿC_xp^x(1 - p)^n-xor p(x : n,p) = ⁿC_xp^x(q)^n-x

Here,

1. n = Number of experiments
2. x = 0, 1, 2, 3, 4, …
3. p = In a single experiment, Probability of success
4. q = In a single experiment, Probability of failure (= 1 – p)

The n-Bernoulli trials are also used in the binomial distribution formula. Here

ⁿC_{x = n!/ x!(n-x)!.Hence,p(x:np) = n!/[x!(n - x)!].p^x.(q)^(n-x)}

For x=2, P(x=2) = ⁵C₂ p² q^5-2 = 5! / 2! 3! × (½)²× (½)³

P(x=2) = 5/16

For x ≥ 4,

p(x ≥ 4) = p(x = 4) + p(x = 5)

Thus, p(x = 4) = ⁵C₄p⁴q^5-4 = 5!/ 4! 1! × (½)⁴ ×(½)¹ = 5 /32

p(x = 5) = ⁵C₅p⁵q^5-5=(½)^5=1/32

Solved example of Binomial Distribution Formula

1. Example: Find the probability of: if a coin is tossed 5 times using binomial distribution
   1. 2 heads exactly
   2. 4 or more heads.
2. Sol: (a) A Bernoulli trial is exemplified by the repeated tossing of a coin. In light of the issue:
   1. Trials conducted: n=5
   2. Probability of head: p= 1/2
   3. Thus, the probability of tail, q =1/2
3. For exactly two heads:
4. (b) 4 or more heads,
5. Answer: Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16
6. Example: For the same question given above, find the probability of getting at most 2 heads.
   1. P(at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)
   2. P(X = 0) = (½)5 = 1/32
   3. P(X=1) = 5C1 (½)5.= 5/32
   4. Answer: Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16

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