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%?