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Binomial Probability Formula is considered to refer to the probability of the exactly x which is supposed to be succeeding towards the n repeated trials as this is deemed to be a part of an experiment which is supposed to have two possible outcomes or implications. These are supposed to be known as the binomial experiment. Therefore, if the probability of the success of an individual trial is considered to be p then such would be having the binomial probability of “nCx⋅px⋅(1−p)n−x .” Thus, the formula is supposed to be “

Binomial distribution is considered to be summarized as the number of trials as well as observations that teach the same probability which is supposed to be attaining the particular value. The binomial distribution for such determines the probability of observing certain specified numbers of the outcomes that are supposed to be successful as these would be coming in specified numbers through the trails. The binomial probability distribution is supposed to be characterized by two parameters and these are supposed to be through the numbers that are independent n and the probability therefore, is considered to be of success p. This particular equation is supposed to be used for obtaining the probability of observing x successes in the N trials as the probability of success is supposed to be based on a single trial as these are denoted by p. In addition to this, the binomial distribution is supposed to help in the assumptions of p which is considered to be p that is fixed for all the trials. Thus, the properties of binomial distribution are considered to have two implications which are true or false or these are considered to be related to failure or yes or no. However, the n number is supposed to be having independent trials as these are supposed to be fixed through n numbers which are on the repeated trials. Therefore, there is a probability of success as well as failure in each of the trial as such varies from equation to equation.

“The binomial distribution formula is for any random variable X, given by; P(x:n,p) = nCx x px (1-p)n-x Or P(x:n,p) = nCx x px (q)n-x” whereas the formula for the binomial distribution is said to be “P(x:n,p) = nC x px (1-p)n-x Or P(x:n,p) = nCx px (q)n-x where,

n = the number of experiments

x = 0, 1, 2, 3, 4, …

p = Probability of success in a single experiment

q = Probability of failure in a single experiment (= 1 – p)

The binomial distribution formula is also written in the form of n-Bernoulli trials, where nCx = n!/x!(n-x)!. Hence, P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x”.

**Example 1:**

“If a coin is tossed 5 times, using binomial distribution find the probability of:

(a) Exactly 2 heads

(b) At least 4 heads.

**Solution:**

(a) The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:

Number of trials: n=5

Probability of head: p= 1/2 and hence the probability of tail, q =1/2

For exactly two heads:

x=2

P(x=2) = 5C2 p2 q5-2 = 5! / 2! 3! × (½)2× (½)3

P(x=2) = 5/16

(b) For at least four heads,

x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)

Hence, P(x = 4) = 5C4 p4 q5-4 = 5!/4! 1! × (½)4× (½)1 = 5/32

P(x = 5) = 5C5 p5 q5-5 = (½)5 = 1/32

Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16.”

**Example 2:**

“For the same question given above, find the probability of getting at most 2 heads.

**Solution:** P(at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)

P(X = 0) = (½)5 = 1/32

P(X=1) = 5C1 (½)5.= 5/32

Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16.”

**Example 3:**

“60% of people who purchase sports cars are men. Find the probability that exactly 7 are men if 10 sports car owners are randomly selected.

**Solution:**

Let’s Identify ‘n’ and ‘X’ from the problem.

The number of sports car owners are randomly selected is n = 10, and

The number to find the probability is X = 7.

**Given:** p = 60%, or 0.6.

Therefore, the probability of failure is q = 1 – 0.6 = 0.4

Now, using the binomial distribution formula

P(x)=n!(n−x)!x!.(p)x(q)n−x=10!(10−7)!7!(0.6)7.t(0.4) 10 - 7 = 120 × 0.0279936 × 0.064 = 0.215

The probability that exactly 7 are men is 0.215 or 21.5%.”

**Ans: **The binomial distribution is supposed to summarize the number of trials and along with such the observations where each of these have the same probability for achieving a particular value and these are supposed to determine the probability by not observing the specified number as such are supposed to be for the successful outcomes or implications which are there for the specified number of trials. The formula is supposed to be “P(X= x) = nCxpxqn-x, where = 0, 1, 2, 3, … P(X = 6) = 105/512. Hence, the probability of getting exactly 6 heads is 105/512.”

**Ans:** The purpose of the binomial formula is considered to be that it permits the individuals to compute the probability by observing the specified number of successes as these are supposed to be during the processes where the things are repeated for a specified number of times.

**Ans: **In the probability theory this is supposed to be discrete and along with such applicable to the events as such is not only considered to have only two possible results but it is supposed to be an experiment that is either successful or it is a failure. Therefore, it can be stated that there are few circumstances where the binomial experiments are supposed to be tossed through head or tail system or through the pass or fail system where the random binomial variable is supposed to be known as binomial probability distribution. The formula is “P(x= 0) = 1/4 = 0.25 P(x= 1) = P(HT) = 1/4 + 1/4 =0.50 P(x =2) = P(HH) = 1/4 = 0.25” for getting heads.

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