Probability Questions

Suppose we roll a fair six-sided die 4 times. What is the probability of getting exactly 2 sixes? Hint: Let X be the number of sixes obtained in the 3 rolls. Then X has a binomial distribution with n = 4 and p = 1/6.

dbinom(2, 4, 1/6)

## [1] 0.1157407

Suppose we were going to flip a biased coin 6 times. The probability of tossing a head is .8 and a tail .2. What is the probability that you’ll toss at most 3 heads.

pbinom(3, 6, .8, lower.tail = TRUE)

## [1] 0.09888

Suppose the probability that a person has blood type B is 0.12. In order to conduct a study concerning people with blood type B, patients are sampled independently of one another until 10 are obtained who have blood type B. Determine the probability that at most 25 patients have to have their blood type determined. Let Y have a negative binomial distribution with p = 0.12 and r = 10.

 pnbinom(15,10,0.12)

## [1] 0.000362934

Refer to question #3, find the probability that exactly 15 patients have to have their blood type determined before the first patient with type B blood is found.

dgeom(15,0.12)

## [1] 0.01763686

A deck of cards contains 20 cards: 7 red cards and 14 black cards. 5 cards are drawn randomly without replacement. What is the probability that exactly 4 red cards are drawn?

dhyper(4,7,14,5)

## [1] 0.02407981

Suppose the number of people that show up at a bus stop is Poisson with a mean of 2.5 per hour, and we want to know the probability that at most 2 people show up in a 4 hour period.

ppois(2, 2.5*4, lower.tail = TRUE)

## [1] 0.002769396

In a standard normal curve, what is the Z-score value whose area below it is 0.30?

qnorm(.30)

## [1] -0.5244005

Find the 97.5th percentile of a normal distribution with mean 5 and standard deviation 1.5.

qnorm(.975,5,1.5)

## [1] 7.939946

Suppose you have a normal distribution with mean 1050 and variance of 1600 and you want to compute the probability that the associated random variable X lessthan 1100.

pnorm(1100,mean=1050,sd=40,lower.tail=TRUE)

## [1] 0.8943502