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 [tex]P(X \le 1 ) = 0.0484[/tex]
The mean is [tex]\mu = 3.2[/tex]
The variance is [tex]\sigma ^2 =2.56[/tex]
The standard deviation is [tex]\sigma =1.6[/tex]
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
[tex]X \~ \ \ \ B(n , p)[/tex]
and the probability distribution function for binomial distribution is
[tex]P(X = x) = ^{n}C_x * p^x * (1- p)^{n-x}[/tex]
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
[tex]P(X \le 1 ) = P(X = 0 ) + P(X = 1 )[/tex]
=> [tex]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} ][/tex]
=> [tex]P(X \le 1 ) = [1 * 1* 0.0352 ] + [15 * 0.20 * 0.0439][/tex]
=> [tex]P(X \le 1 ) = 0.0484[/tex]
Generally the mean for this experiment is mathematically represented as
[tex]\mu = n * p[/tex]
=> [tex]\mu = 16 * 0.2[/tex]
=> [tex]\mu = 3.2[/tex]
Generally the variance is mathematically represented as
[tex]\sigma ^2 = n * p * (1- p )[/tex]
=> [tex]\sigma ^2 = 16 * 0.2 * (1- 0.2)[/tex]
=> [tex]\sigma ^2 =2.56[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{\sigma ^2 }[/tex]
=> [tex]\sigma =1.6[/tex]
