Some Sample Questions



1.  Select a stock at random.  The mean return for stocks is .11 (let's say), with SD of 0.28.  Assume these returns are normally distributed.

a. If I select a stock at random, what's the probability I will lose money?
b. If I randomly select 9 stocks, what's the probability that my average returns will be less than zero?
c. If I randomly select 25 stocks, what's the probability I'll lose money?

2. Two fair, six-sided dice are tossed.  If the sum of the dice is 7, you win $10.  If they don't add to 7, you lose x dollars, where x is to be determined.
a. What's the probability that you will win $10?  (Hint, the total number of outcomes is 36.  How many of these add to 7?)
b.  How much should x be to make the game "fair"?
c. Suppose x = $3.  What are the mean winnings?  What's the SD of your winnings?
d. If you played the game 16 times, each time betting $3, how much would you expect your winnings to be?  Would would be the SD of your winnings?
e.  If you played the game 16 times, each time betting $3, how much woul dyou expect your average winnings to be?  What would be the Standard Error for your average winnings?  (Standard Error (SE) is the SD of the average)>
f.  What's the approximate probability that you will win money playing this game 16 times?

3.  The Central Limit Theorem can also be used for coin tosses.  Suppose you toss a coin n times, and the probability that this coin comes up "heads" is p.
(If the coin is "fair", then p  = 1/2.)  The CLT theorem then says that if X represents the number of heads, then for n sufficiently large, X has an approximately normal distribution with mean np and SD sqrt(np(1-p).

a. Toss a fair coin 100 times.  What's the approximate probability of getting between 45 and 65 heads?
b. Toss a fair coin 100 times. What's the approximate probability of getting more than 70 heads?

The CLT also says that the random variable (X/n)  (the percentage of flips that came up heads) follows an approximate normal distribution with mean p and
SD sqrt( p*(1-p)/n).

c. What's the approximate probability that you will get  less than 60% heads?  Less than 40%?
 
 

Answers will be posted on Tuesday.