Section 5.1, p. 238

5,7

Section 10.2 (p. 504)
4abc, 5a
A. In a simple dice game, a die is rolled.  You win $1 for each spot.
a) What is your expected winning?
b) Would you pay $4 to play this game?  Why or why not?
c) How much would you spend to make this a "fair" game?  A fair game is one in which the expected value of the player and the "house" (the casino) is equal.

B. Suppose that on one particular day the lottery  sold tickets with the payoffs as below. A ticket costs $1.  (So if the "payoff" is 0, you actually just lost 1 dollar, and if the payoff is 1, then you broke even.)
 
 

payoff	probability
0	.5
$1	.25
$5	.10
$10	.10
$20	.05

a) What's the expected payoff?
b) True or false and explain:  Everyone who played the lottery won the expected amount.
c) True or false and explain: if we averaged the winnings of 100 randomly selected players, the average would equal the expected amount.

C. Toss a fair coin 4 times.
a) Write the sample space.  (e.g HHT Tis one outcome).  Hint: there should be 16 outcomes.
b) Let X represent the number of heads.  Use your sample space listing to find the pdf of X.
c) What is the probability of getting 2 or more heads?
d) What is the probability of getting at least one head?
e) What is the probability of getting all tails?  Add your answer to this question to (d).  Why does the sum equal 1?  (If it doesn't, check your work!)
f) Use the pdf to find the expected number of heads.
g) Because this is a binomial experiment, the expected number of heads is n*p.  Find n*p and make sure you get the same answer as in (f).

D.  There is some evidence that the true probability that a child will be a girl is closer to 0.509 than to .500.  Assume it is .509.  We're interested in knowing the expected number of girls in a family of 3 children.
a) Go through the four conditions of a binomial experiment and explain why the binomial model is a good one for this problem.
b) Calculate the expected number of girls in a family of 3.  Compare this to the expected value if the true probability of a girl was .50.

E. For each of the following random variables, say whether or not it is a binomial random variable, and explain why.  If it is, find the expected value.
a) 40% of people in a large city believe that energy conservation should be a vital part of a national energy policy.  A survey samples 1000 people, selected randomly with replacement and wants to count the number that will say they agree that energy conservation should be a vital part of a national energy policy.
b) Same problem, but now the survey selections 1000 people randomly withOUT replacement.  (This is called a Simple Random Sample.)
c) 4 cards are dealt off of a standard deck of playing cards.  The deck has been well-shuffled.  We're interested in counting the number of aces.  (There are 4 aces in a deck of 52 cards.)
d) A multiple choice quiz has 5 choices for each answer.  There are 50 questions on the quiz.  We're interested in counting the number of correct answers if a person guesses randomly.
e) A one-mile stretch of Santa Monica Blvd has 10 traffic lights.  Suppose that each light is red 35% of the time.  We're interested in the number of green lights we will hit as we drive along this stretch.
 
 

F. Find the standard deviation for X in Problem A.
G. Find the standard deviation for your winnings in B.
H. Find the standard deviation for the number of heads in C.
I. Find the standard deviation for the number of girls in D.  (Assuming p = 0.509.)