MATH 1550H Final: MATH1550H-Final (5)
Mathematics 1550H – Introduction to probability
Trent University, Summer 2014
Final Examination
Friday, 6 August, 2014
Time: 3 hours Brought to you by Stefan Blank.
Instructions: Do both of parts ♥
♥
♥and ♦
♦
♦, and, if you wish, part ♣. Show all your work
and simplify answers as much as practicable. If in doubt about something, ask!
Aids: Calculator; one 8.5′′ ×11′′ or A4 aid sheet; standard normal table; ≤1010! neurons.
Part ♥
♥
♥.Do all of 1–5.[Subtotal = 70/100]
1. A fair standard six-sided die is rolled four times. Let Xbe the number of times that
odd-numbered faces come up, and let Ybe the sum of the odd-numbered faces that
come up.
a. What is the probability function of X?[5]
b. Compute E(X) and Var(X). [5]
c. Without computing it from scratch, what should E(Y) be? Explain why. [5]
2. A continuous random variable Zis uniformly distributed between 0 and 3.
a. What is the expected value of Z2?[10]
b. What is the probability that Z2is at most 8? [5]
3. A hand of five cards is drawn simultaneously (i.e. without order or replacement) from
a standard 52-card deck.
a. What is the probability that the hand is a “full house,” that is, that it includes
a pair of one kind and a triple of another kind? [8]
b. What is the probability that all the cards in the hand are of different kinds? [7]
4. Suppose Yis a normally distributed continuous random variable with expected value
µ= 3 and standard deviation σ= 2.
a. Compute P(Y≥0) using your standard normal table. [5]
b. Find the median of Y,i.e. the number msuch that P(Y≤m) = 0.5, with the
help of your standard normal table. [5]
5. A fair coin is tossed, and then tossed some more until it comes up again with whatever
face came up on the first toss.
a. What are the sample space and probability function? [7]
b. Let Abe the event that no more than three tosses took place and let Bbe the
event that the second toss was a head. Determine whether the events Aand B
are independent or not. [8]
[Parts ♦
♦
♦and ♣are on page 2.]
1
Document Summary
Instructions: do both of parts and , and, if you wish, part . Show all your work and simplify answers as much as practicable. Aids: calculator; one 8. 5 11 or a4 aid sheet; standard normal table; 1010! neurons. [subtotal = 70/100: a fair standard six-sided die is rolled four times. [7: suppose y is a normally distributed continuous random variable with expected value. = 3 and standard deviation = 2: compute p (y 0) using your standard normal table. [5: find the median of y , i. e. the number m such that p (y m) = 0. 5, with the help of your standard normal table. [7: let a be the event that no more than three tosses took place and let b be the event that the second toss was a head. Determine whether the events a and b are independent or not. [parts and are on page 2. ]
