In: Statistics and Probability

a die is rolled 6 times let X denote the number of 2's that appear on the die.

1. show that X is binomial.

2. what is the porbaility of getting at least one 2.

3. find the mean and the standard deviaion of X

Solution:

1)

A die is rolled 6 times.

Number of trials = n = 6

let X denote the number of 2's that appear on the die.

For the event "getting 2 " ,

probability of success = p = 1/6

probability of failures = q = 1 - p = 5/6

Trials are independent.

Probability of success is constant from trial to trial.

So , X is binomial with n = 6 , p = 1/6

2)

Using binomial probability formula ,

P(X = x) = (_{n} C _{x}) * p^{x} * (1 -
p)^{n - x} ; x = 0 ,1 , 2 , ....., n

Now ,

P(at least one 2)

= P(X 1)

= 1 - { P(X < 1) }

= 1 - { P(X = 0) }

= 1 - { (_{6} C _{0}) * (1/6)^{0} *
(5/6)^{6 - 0} }

= 1 - { 0.33489797668 }

= 0.6651

**P(at least one 2) = 0.6651**

3)

Mean = = n * p = 6 * (1/6) = 1

Standard deviation = = n * p * q = [6 * (1/6) * (5/6)] = 0.9129

**Mean = 1 **

**Standard deviation = 0.9129**

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