Combining Probabilities and The Law of Large Numbers

1. Suppose there are 15 jelly beans in a box 2 red, 3 blue, 4 white, and 6 green. A jelly bean is selected at random.
a) What is the probability that the jelly bean is white? _____
b) What is the probability that the jelly bean is not white? _____
c) What is the probability that the jelly bean is green? _____
d) What is the probability that the jelly bean is red or green? _____
e) What is the probability that the jelly bean is neither red nor green? _____

2. The probability of a $2 winner in a particular state lottery is 1 in 10, the probability of a $5 winner is 1 in 50, and the probability of a $10 winner is 1 in 500.
a) What is the probability of getting either a $2, or $5, or $10 winner? _____
b) What is the probability of getting a $2 winner? _____
c) If you buy 50 lottery tickets, what is the probability that you will get at least one $5 winner? _____
d) If you buy 500 lottery tickets, what is the probability that you will get at least one $10 winner?
_____

3. A new cold medication was tested by giving 125 people the drug and 100 people a placebo. A control group consisted of 115 people who were given no treatment. The number of people in each group who showed improvement is shown in the table below.
Cold Drug Placebo Control Total
Improvement 72 55 38 165
No Improvement 53 45 77 175
Total 125 100 115 340

a) What is the probability that a randomly selected person in the study either was given the placebo or was in the control group? ____________

b) What is the probability that a randomly selected person either was given the drug or improved? _____________

c) What is the probability that a randomly selected person was given the drug and improved? ______________

d) What is the probability that a randomly selected person who improved was given the drug? ________

e) What is the probability that a randomly selected person who was given the drug improved? ________

f) Based on these data, does the drug appear to be effective? ____________

Expected Value - The theoretical value for you, the player in a game, is calculated as follows: If an outcome results in a win for you, multiply its probability by the profit you win. If an outcome results in a loss for you, multiply the probability by the negative of the amount you lose. Your expected value is the sum of all these positive and negative numbers. A simplified formula for this is
Expected value = P(Win) x Profit - P(Lose) x Loss
The expected value is a measure of the average amount you can expect to win (or lose) per play in the long run.
If the expected value is zero, the game is fair.

4. A standard roulette wheel has numbers 1 through 36 alternately colored red and black. It also has a green 0 and a green 00 called "double zero." The wheel is spun with a small white ball inside. When the wheel stops, the ball falls into a numbered slot, which determines the winners. Successive spins of a wheel yield independent results. There are many wagers you can place at the roulette table. For example, you may bet on the color, or on whether the number is odd or even, or on a single number. Note that, even though 0 is in fact an even number, in roulette both 0 and 00 count an neither odd nor even.
a) You are playing roulette, and you bet a dollar on the number 10. If 10 comes up you win $35(profit). If anything else comes up, you lose your dollar. What is your expected value for this bet? ______________

b) If you bet $1 on an even number, you win $1 (profit) if any of the even numbers 2 through 36 come up, and you lose your dollar if 0, 00, or any odd number between 1 and 35 comes up. What is your expected value for this wager? ________________

5. Suppose a company charges a premium of $150 per year for an insurance policy for storm damage to roofs. Actuarial studies show that in case of a storm, the insurance company will pay out an average of $8000 for damage to a composition shingle roof and an average of $12,000 for damage to a shake roof. They also determine that out of every 10,000 policies, there are 7 claims per year made on composition shingle roofs and 11 claims per year made on shake roofs. What is the company's expected value (i.e., expected profit) per year of a storm insurance policy? What annual profit can the company expect if it issues 1000 such policies?
Determine the probability of a composition shingle roof claim out of 10,000 = ______
Determine the probability of a shake roof claim out of 10,000 = ______
How many claims are made out of 10,000? = _______
What is the probability of no claims out of 10,000? = _______
How much does each shingle roof claim cost the company, don't forget each person pays $150 for the insurance? = ______
How much does each shake roof claim cost the company? = _____
How much money does the company make from each customer that does not make a claim? = ______
Calculate the Expected Value (this time you have 2 values to subtract from the Profit in the formula.
Write formula used here_______________________________________________________
What is the company's Expected Value = _________
How much profits will the company makes if it issues 1000 such policies? = _______