Stats / Probability?

Question

Are your finances, buying habits, medical records, and phone calls really private? A real concern for many adults is that computers and the internet are reducing privacy. A survey conducted and according to the survey, 37% of adults are concerned that employers are monitoring phone calls. Use the binominal distribution formula to calculate the probability that

a. Out of 5 adults, none is concerned that employers are monitoring phone calls.

b. B. out of 5 adults, all are concerned that employers are monitoring phone calls.

c. Out of 5 adults. Exactly 3 are concerned that employers are monitoring phone calls.

Answer 1

B.

Minitab

Answer 2

Let X be the number of adults concerned that employers are monitoring phone calls.  X has the binomial distribution with n = 5 trials and success probability p = 0.37



In general, if X has the binomial distribution with n trials and a success probability of p then

P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)

for values of x = 0, 1, 2, ..., n

P[X = x] = 0 for any other value of x.

The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.

Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.

X ~ Binomial( n = 5 , p = 0.37 )

the mean of the binomial distribution is n * p = 1.85

the variance of the binomial distribution is n * p * (1 - p) = 1.1655

the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 1.079583

The Probability Mass Function, PMF,

f(X) = P(X = x) is:

P( X =  0 ) =  0.09924365 <<<< None concerned

P( X =  1 ) =  0.2914298

P( X =  2 ) =  0.3423143

P( X =  3 ) =  0.2010418 <<<< exactly 3 are concerned

P( X =  4 ) =  0.05903607

P( X =  5 ) =  0.006934396 <<<< All are concerned