List the four requirements for a binomial distribution Descr

List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements.

Solution

The four requirements are:

Firstly, examples which are not binomial :

now, let\'s consider an example of binomial distribution:

Question. : Toss a coin for 12 times. What is the probability of getting exactly 7 heads.

here,

- fixed number of trials(n) = 12

- Each new trial is independent of the previous trial.

- there are only two outcomes possible(head / tail)

- On each trial P(success) = 0.5 and P(failure) = 0.5 i.e. constant.

solution:

Step 1:

Here, Number of trials n = 12 Number of success r = 7 since we define getting a head as success Probability of success on any single trial p = 0.5

Step 2:

To Calculate _nC_r formula is used. _nC_r = ( n! / (n-r)! ) / r! = ( 12! / (12-7)! ) / 7! = ( 12! / 5! ) / 7! = ( 479001600 / 120 ) / 5040 = ( 3991680 / 5040 ) = 792

Step 3:

Find p^r. p^r = 0.5⁷ = 0.0078125

Step 4:

To Find (1-p)^n-r Calculate 1-p and n-r. 1-p = 1-0.5 = 0.5 n-r = 12-7 = 5

Step 5:

Find (1-p)^n-r. = 0.5⁵ = 0.03125

Step 6:

Solve P(X = r) = _nC_r p ^r (1-p)^n-r = 792 * 0.0078125 * 0.03125 = 0.193359375 The probability of getting exactly 7 heads is 0.19

The probability of getting exactly x success in n trials, with the probability of success on a single trial being p is: