Question

if 40% of all commuters ride to work in carpools, find the probability that if eight commuters are selected, five of them will ride in carpools.

Answer 1

do you still need help?

Answer 2

yes i do

Answer 3

okay

Answer 4

Let X be the number of commuters who carpool. X has the binomial distribution with n = 8 trials and success probability p = 0.4

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 = 8 , p = 0.4 )

the mean of the binomial distribution is n * p = 3.2
the variance of the binomial distribution is n * p * (1 - p) = 1.92
the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 1.385641

The Probability Mass Function, PMF,
f(X) = P(X = x) is:

P( X = 0 ) = 0.01679616
P( X = 1 ) = 0.08957952
P( X = 2 ) = 0.2090189
P( X = 3 ) = 0.2786918
P( X = 4 ) = 0.2322432
P( X = 5 ) = 0.1238630
P( X = 6 ) = 0.04128768
P( X = 7 ) = 0.00786432
P( X = 8 ) = 0.00065536