Binomial probability formula

Binomial is deals with the polynomial that has two terms is known as a binomial e.g.5x + 3y, 2x^2 – 5xy.Probability is the arithmetical quantify of the possibility of an event to occur. If in an examine there are n feasible ways completely and mutually exclusive and out of them in m ways in the event A occur, it is given by (m / n) if in a random sequence of n trails of an events, M are favor to the event, then ratio is (M / n).by studying in both combined as binomial probability as P(A) = m / n.

The possibility of an event can be conveyed as a binomial probability if its conclusions can be wrecked down into two probability of p and q, where p and q are balancing (i.e. p + q = 1)


Binomial Probability Formula - Explained


The probability of getting exact value of k in n trials is given by using the formula,

`P(X = x) = ((n),(k)) p^k q^(n-k) `
where
` q= (1-p)`
or
`P(X = r) =nCr p^r(1-p)^(n-r)`

where
n = Number of events.
r = Number of successful events in the trials.
p = Probability of success in the single trial of the events.
nCr = `(( (n!) / (n-r)! ) / (r!))`
1-p = Probability of failure in the trial.


Examples on Binomial Probability Distribution

Example 1:

Assume that there are taking about 10 question multiple choice test. If each question has four choices and you have to guess on each question, what is the probability of getting exactly 7 questions correct using binomial Probability formula?


Solution:
General Formula for finding solution is,

`P(X = x) = ((n),(k)) p^k q^(n-k) `

n = 10
k = 7
n – k = 3
p = 0.25 = probability of guess the correct answer on a question
q = 0.75 = probability of guess the wrong answer on a question

P(7 Correct guesses out of 10 questions) = `((10),(7))` (0.25)7(0.75)3
? 0.0031 approximate value

Therefore if someone guesses 10 answers on a multiple choice test with 4 options, they have about a 5.8% chance of getting 5 and only 5 correct answers. If 5 or more correct answers are needed to pass then probability of passing can be calculated by adding the probability of getting 5 (and only 5) answers correct, 6 (and only 6) answers correct, and so on up to 10 answers correct. Total probability of  5 or more correct answer is approximate percentage is  7.8


Example 2:
Suppose a die is tossed 7 times. What is the probability of getting exactly 3 fours?

Solution:

General Formula for finding solution

`P(X = r) =nCr p^r(1-p)^(n-r)`

Step 1:
Number of trials n = 7
Number of success r =3

Probability of success in any single trial p is given as 1/6 or 0.167

Step 2:

To calculate nCr formula is used.

nCr = `(n!)/((n-r)!(r!))`

= `(7!)/((7-3)!(3!))`

= `(7!)/((4)!(3!))`

= `(5040)/((24)(6))`

= `(5040)/(144)`

= 35

Step 3:

Find pr.
pr =  0.1673
= 0.004657463

step 4:

To Find (1-p)n-r Calculate 1-p and n-r.
1-p = 1-0.167 = 0.833
n-r = 7-3 = 4

Step 5:

Find (1-p)n-r.
= 0.8334 = 0.481481944

Step 6:

Solve P(X = r) = nCr p r (1-p)n-r
= 10 × 0.004657463 × 0.481481944
=  0.0224248434

The probability of getting exactly 3 fours is 0.0224248434