A college Statistic question?

Question

A solitaire player wins 65% of her games. If she plays four games what is the probability that she loses at least one game?

Answer 1

35%

Answer 2

Let X be the number of games lost.  X has the binomial distribution with n = 4 trials and success probability p = 1 - 0.65 = 0.35



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 = 0.35 )

the mean of the binomial distribution is n * p = 1.4

the variance of the binomial distribution is n * p * (1 - p) = 0.91

the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 0.9539392

The Probability Mass Function, PMF,

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

P( X =  0 ) =  0.1785063

P( X =  1 ) =  0.384475

P( X =  2 ) =  0.3105375

P( X =  3 ) =  0.111475

P( X =  4 ) =  0.01500625

The Cumulative Distribution Function, CDF,

F(X) = P(X ≤ x) is:

x

∑ P(X = t) =

t = 0

P( X ≤ 0 ) =  0.1785063

P( X ≤ 1 ) =  0.5629812

P( X ≤ 2 ) =  0.8735187

P( X ≤ 3 ) =  0.9849937

P( X ≤ 4 ) =  1

1 - F(X) is:

n

∑ P(X = t) =

t = x

P( X ≥ 0 ) =  1

P( X ≥ 1 ) =  0.8214937 = 1 - P(X = 0) <<<< ANSWER

P( X ≥ 2 ) =  0.4370188

P( X ≥ 3 ) =  0.1264813

P( X ≥ 4 ) =  0.01500625