Binomial Distribution Formula

The binomial distribution formula helps to check the probability of getting “x” successes in “n” independent trials of a binomial experiment. To recall, the binomial distribution is a type of probability distribution in statistics that has two possible outcomes. In probability theory, the binomial distribution comes with two parameters n and p.

The probability distribution becomes a binomial probability distribution when it meets the following requirements.

1. Each trial can have only two outcomes or the outcomes that can be reduced to two outcomes. These outcomes can be either a success or a failure.
2. The trails must be a fixed number.
3. The outcome of each trial must be independent of each others.
4. And the success of probability must remain the same for each trial.

Binomial Distribution Formula in Probability

The formula for the binomial probability distribution is as stated below:

Where,

• n = Total number of events
• r (or) x = Total number of successful events.
• p = Probability of success on a single trial.
• ⁿC_r = [n!/r!(n−r)]!
• 1 – p = Probability of failure.

Try This: Binomial Distribution Calculator

Examples on Binomial Distribution Formula

Example 1:

A coin is tossed12 times. What is the probability of getting exactly 7 heads?

Solution:

Given that a coin is tossed 12 times. (i.e) n= 12

Thus, a probability pf gettig head in single toss = ½. (i.e) p = ½.

So, 1-p = 1-½ = ½.

We know that the binomial probability distribution is P(r) = ⁿC^r · p^r (1 − p)^n−r.

Now, we have to find the probability of getting exactly 7 heads.(i.e) r = 7.

Substituting the values in the binomial distribution formula, we get

P(7) = ¹²C₇ · (½)⁷ (½)¹²⁻⁷

P(7) =  792· (½)⁷ (½)⁵

P(7) = 792.(½)¹²

P(7) = 792 (1/4096)

P(7) = 0.193

Therefore, the probability of getting exactly 7 heads is 0.193.

Example 2: 

A coin that is fair in nature is tossed n number of times. The probability of the occurrence of a head six times is the same as the probability that a head comes 8 times, then find the value of n.

Solution:

The probability that head occurs 6 times = ⁿC₆ (½)⁶ (½)^n-6

Similarly, the probability that head occurs 8 times = ⁿC₈ (½)⁸ (½)^n-8

Given that, the probability of the occurrence of a head six times is the same as the probability that a head comes 8 times,

(i.e) ⁿC₆ (½)⁶ (½)^n-6 = ⁿC₈ (½)⁸ (½)^n-8

⇒ⁿC₆(½)ⁿ =  ⁿC₈ (½)ⁿ

⇒ⁿC₆ =  ⁿC₈

⇒ 6 = n-8

⇒ n= 14.

Therefore, the value of n is 14.

Example 3:

The probability that a person can achieve a target is 3/4. The count of tries is 5. What is the probability that he will attain the target at least thrice?

Solution:

Given that, p = ¾, q = ¼, n = 5.

Using binomial distribution formula, we get P(X) = ⁿC_x · p^x (1 − p)^n−x

Thus, the required probability is: P(X = 3) + P(X=4) + P(X=5)

=  ⁵C₃ · (¾)³ (¼ )² + ⁵C₄ · (¾)⁴ (¼ )¹ +⁵C₅ · (¾)⁵ 

= 459/512.

Therefore, the probability that the person will attain the target atleast thrice is 459/512.

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