Question

All three ASAP Statistics and probability.

A multiple-choice test contains 15 questions, each with four answers: A, B, C, and D. Assume thata student randomly guesses on each question.

•What is the probability that a student less than 2 questions correctly?

•What is the probability that the 5th question is the first question the student gets right?

•How many questions do you expect the student to get correct?

2. Astronomers treat the number of stars in a given volume of space as a Poisson random variable.The density in the Milky Way Galaxy in the vicinity of our solar system is ten stars per 16 cubiclight-years.

•What is the probability of at least one star in 5 cubic light-years?

•How many stars would you expect there to be in 50 cubic light-years?

3. Consider PDFf(x) =kcos(x) for−π/2≤X≤π/2

.•Find kso thatf(x) is a PDF

•FindP(−π/6< X≤π/4)

Answer 1

1)

Let X be a binomial random variable which denotes the number of questions the student guesses correctly

where n = 15, p = 0.25

Probability that a student get less than 2 questions correctly = P(X < 2)

= P(X = 1) + P(X = 0) =

= 0.08

Probability that the 5th question is the first question the student gets right = = 0.079

Number of question you would expect the student to get correct = np = 3.75

2)

= 10 stars per 16 cubiclight-years

= 1/16 stars per cubiclight-years

Average number of stars in 5 cubic light-years = 5/16

Probability of atleast one star in 5 cubic light-years = = 0.2684

Number of stars expected to be there in 50 cubic light-years = 50*5/16 = 15.625

3)

f(x) = kcos(x)

For f(x) to be a valid PDF,

-> k = 1/2

= = 0.6036