Question

For each of the following binomial random variables, specify n and p. (a) A fair die is rolled 50 times. X = number of times a 5 is rolled. n = 50 Correct: Your answer is correct. p = 1/5 Incorrect: Your answer is incorrect. (b) A company puts a game card in each box of cereal and 1/100 of them are winners. You buy sixteen boxes of cereal, and X = number of times you win. n = 16 Correct: Your answer is correct. p = 1/100 Correct: Your answer is correct. (c) Jack likes to play computer solitaire and wins about 25% of the time. X = number of games he wins out of his next 26 games.μ =

Answer 1

a)
`n = 50, p = (1)/(6)`

b)
`n = 16, p = (1)/(100)`

c)
`n = 26, p = 0.25, \mu = 6.5`

Explanation:

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinatios of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And
`p` is the probability of X happening.

(a) A fair die is rolled 50 times. X = number of times a 5 is rolled

The die is rolled 50 times, so
`n = 50`.

Each roll can have 6 outcomes. So the probability that 5 is rolled is
`p = (1)/(6)`

(b) A company puts a game card in each box of cereal and 1/100 of them are winners. You buy sixteen boxes of cereal, and X = number of times you win.

You buy 16 boxes of cereal, so
`n = 16`.

1 of 100 are winners. So
`p = (1)/(100)`.

(c) Jack likes to play computer solitaire and wins about 25% of the time. X = number of games he wins out of his next 26 games.

He plays 26 games, so
`n = 26`.

He wins 25% of the time, so
`p = 0.25`

We have that
`\mu = np`. So
`\mu = 26*0.25 = 6.5`