Please use Binomial distribution to compute the probability

Please use Binomial distribution to compute the probability of obtaining at least one \"Six\" in rolling a fair die 4 times.

Solution

Let X be the number of 6\'s rolled in four rolls of a fair die. X has the binomial distribution with n = 4 trials and success probability p = 1/6 = 0.1666667

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 = 4 , p = 0.1666667 )

the mean of the binomial distribution is n * p = 0.6666667
the variance of the binomial distribution is n * p * (1 - p) = 0.5555556
the standard deviation is the square root of the variance = ( n * p * (1 - p)) = 0.745356

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

P( X = 0 ) = 0.4822531
P( X = 1 ) = 0.3858025
P( X = 2 ) = 0.1157407
P( X = 3 ) = 0.0154321
P( X = 4 ) = 0.000771605


The Cumulative Distribution Function, CDF,
F(X) = P(X x) is:

x
P(X = t) =
t = 0

P( X 0 ) = 0.4822531
P( X 1 ) = 0.8680556
P( X 2 ) = 0.9837963
P( X 3 ) = 0.9992284
P( X 4 ) = 1


1 - F(X) is:

n
P(X = t) =
t = x

P( X 0 ) = 1
P( X 1 ) = 0.5177469 = 1 - P(X = 0) <<<< ANSWER
P( X 2 ) = 0.1319444
P( X 3 ) = 0.01620370
P( X 4 ) = 0.000771605