The binomial distribution is a type of probability distribution in statistics that has two possible outcomes. In probability theory, the binomial distribution come with two parameters n and p.
The probability distribution becomes a binomial probability distribution when it meets the following requirements.
- Each trail 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.
- The trails must be a fixed number.
- The outcome of each trail must be independent of each other.
- And the success of probability must remain the same for each trail.
The formula for binomial probability is as stated below:
\[\large P(x) = \frac{n!}{r!(n-r)!} . p^{r}(1-p)^{n} = C(n, r).p^{r}(1-p)^{n-r}\]
Where,
n = Total number of events
r = Total number of successful events.
p = Probability of success on a single trial.
n = Total number of events
r = Total number of successful events.
p = Probability of success on a single trial.
nCr = $\frac{n!}{r!(n − r)!}$
1 – p = Probability of failure.
Solved Examples
Question 1:
Toss a coin for 12 times. What is the probability of getting exactly 7 heads.
Solution:
Toss a coin for 12 times. What is the probability of getting exactly 7 heads.
Solution:
Number of trails (n) = 12
Number of success (r) =Â 7
probability of single trail(p) = $\frac{1}{2}$ = 0.5
Number of success (r) =Â 7
probability of single trail(p) = $\frac{1}{2}$ = 0.5
nCr = $\frac{n!}{r!(n – r)!}$
= $\frac{12!}{7!(12 – 7)!}$
= $\frac{12!}{7!5!}$
= $\frac{95040}{120}$
=Â 792
pr = $0.5^{7}$ = 0.0078125
To Find $(1-p)^{n-r }$Â Calculate 1-p and n-r.
1-p = 1-0.5 = 0.5
n-r = 12-7 = 5
1-p = 1-0.5 = 0.5
n-r = 12-7 = 5
$(1-p)^{n-r }$ = $0.5^{7}$ =Â 0.03125
Solve P(X = r) = nCr. pr . $(1-p)^{n-r }$
= 792 x 0.0078125 x 0.03125
= 0.193359375
The probability of getting exactly 7 heads is 0.19
