What is the probability that Martha will fly to Las Vegas exactly twice?
December 26, 2009 by Lisa McLaughlin
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Martha is a flight attendant stationed in Denver. Her assignments are rotated randomly between flights to Dallas; Seattle; Portland; and Las Vegas. Consider a success being a trip to Las Vegas.
What is the probability that Martha will fly to Las Vegas exactly twice?
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Let X be the number of flight Martha has to vegas. X has the binomial distribution with n = 4 trials and success probability p = 0.25
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 , p )
the mean of the binomial distribution is n * p = 1
the variance of the binomial distribution is n * p * (1 – p) = 0.75
the standard deviation is the square root of the variance = √ ( n * p * (1 – p)) = 0.8660254
The Probability Mass Function, PMF,
f(X) = P(X = x) is:
P( X = 0 ) = 0.3164063
P( X = 1 ) = 0.421875
P( X = 2 ) = 0.2109375 ← answer
P( X = 3 ) = 0.046875
P( X = 4 ) = 0.00390625
I assume you mean that she is flying 2 assignments, what is the probability that they are both to Vegas. Otherwise, you don’t have enough information. There are 4 destinations, so the odds of going to any 1 destination is 1/4. The odds of going to that destination twice is (1/4)(1/4) = 1/16
1/4 * 1/4 = 1/16