Question

Determine the probability p(1 or fewer) for a binomial experiment with trials and the success probability . Then find the mean, variance, and standard deviation.

Answer 1

Complete Question

Determine the probability P(1 or fewer) for a binomial experiment with n = 16 trials and the success probability p = 0.2. Then find the mean, variance, and standard deviation

Answer:

The probability is $P(X \le 1 ) = 0.0484$

The mean is $\mu = 3.2$

The variance is $\sigma ^2 =2.56$

The standard deviation is  $\sigma =1.6$

Step-by-step explanation:

From the question we are told that

    The probability of success is  p = 0.2

     The sample size is  n =  16

Generally this experiment follows a binomial distribution  

i.e  

         $X \~ \ \ \ B(n , p)$

and the probability distribution function for binomial  distribution is  

      $P(X = x) = ^{n}C_x * p^x * (1- p)^{n-x}$

Here C stands for combination hence we are going to be making use of the combination function in our calculators  

Generally the the probability P(1 or fewer) is mathematically represented as

       $P(X \le 1 ) = P(X = 0 ) + P(X = 1 )$

=>   $P(X \le 1 ) = [^{15}C_0 * (0.2)^0 * (1- 0.2)^{15-0}] + [^{15}C_1 * (0.20)^1 * (1- 0.20)^{15-1} ]$

=>   $P(X \le 1 ) = [1 * 1* 0.0352 ] + [15 * 0.20 * 0.0439]$

=>   $P(X \le 1 ) = 0.0484$

Generally the mean for this experiment is mathematically represented as

     $\mu = n * p$

=>  $\mu = 16 * 0.2$

=>  $\mu = 3.2$

Generally the variance is mathematically represented as

     $\sigma ^2 = n * p * (1- p )$

=>  $\sigma ^2 = 16 * 0.2 * (1- 0.2)$

=>  $\sigma ^2 =2.56$

Generally the standard deviation is mathematically represented as

     $\sigma = \sqrt{\sigma ^2 }$

=>    $\sigma =1.6$