Answer:
a) ![n = 50, p = \frac{1}{6}](https://tex.z-dn.net/?f=n%20%3D%2050%2C%20p%20%3D%20%5Cfrac%7B1%7D%7B6%7D)
b) ![n = 16, p = \frac{1}{100}](https://tex.z-dn.net/?f=n%20%3D%2016%2C%20p%20%3D%20%5Cfrac%7B1%7D%7B100%7D)
c) ![n = 26, p = 0.25, \mu = 6.5](https://tex.z-dn.net/?f=n%20%3D%2026%2C%20p%20%3D%200.25%2C%20%5Cmu%20%3D%206.5)
Step-by-step 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}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
In which
is the number of different combinatios of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
And
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
.
Each roll can have 6 outcomes. So the probability that 5 is rolled is ![p = \frac{1}{6}](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7B1%7D%7B6%7D)
(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
.
1 of 100 are winners. So
.
(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
.
He wins 25% of the time, so ![p = 0.25](https://tex.z-dn.net/?f=p%20%3D%200.25)
We have that
. So ![\mu = 26*0.25 = 6.5](https://tex.z-dn.net/?f=%5Cmu%20%3D%2026%2A0.25%20%3D%206.5)