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Pick a card, any card. You draw a card from a deck. If you get a red card, you win nothing. If you get a spade, you win $5. For any club, you win $10 plus an extra $20 for the ace of clubs.
a) Create a probability model for the amount you win at this game.
b) Find the expected amount youll win.
c) How much would you be willing to play this game?
Software. A small software company bids on two contracts. It anticipates a profit of $50,000 if it gets the larger contract and a profit of $20,000 on the smaller contract. The company estimates theres a 30% chance it will get the larger contract and a 60% it will get the smaller contract. Assuming the contracts will be awarded independently, whats the expected profit?
Red lights. A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. She estimates the probability model for the number of red lights she hits, as shown below.
X= # of red012345P(X=x)0.050.250.350.150.150.05a) How many red lights should she expect to hit each day?
b) Whats the standard deviation?
Defects. A consumer organization inspecting new cars found that many had appearance defects (dents, scratches, paint chips, etc.). While none had more than three of these defects, 7% had three, 11% had two, and 21% had one defect. Find the expected number of appearance defects in a new car and the standard deviation.
You bet! You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose.
a) Create a probability model (distribution) for the amount you win at this game.
b) Find the expected amount you will win.
c) How much are you willing to pay to play this game?
Carnival. A carnival game offers a $100 cash prize for anyone who can break a balloon by throwing a dart at it. It costs $5 to play, and you are willing to spend up to $20 trying to win. You estimate that you have about a 10% chance of hitting the balloon on any throw.
a) Create a probability distribution for this carnival game.
b) Find the expected number of darts you will throw.
c) Find your expected winnings.
Fire! An insurance company estimates that it should make an annual profit of $150 on each homeowners policy written, with a standard deviation of $6000.
a) Why is the standard deviation so large?
b) If it writes only two of these policies, what are the mean and standard deviation of the annual profit?
c) If it writes 10,000 of these policies, what are the mean and standard deviation of the annual profit?
d) Do you think the company is likely to make a profit? Explain.
e) What assumptions underlie your analysis? Can you think of circumstances under which those assumptions might be violated? Explain.
Medley. In the 4 x 100 medley relay event, four swimmers swim 100 yards, each using a different stroke. The swim team for the USA Olympic team is preparing for the gold medal round and looks at the times their swimmers have posted and creates a model based on the following assumptions:
The swimmers performances are independent.
Each swimmers times follow a normal model.
The means and standard deviations of the times (in seconds) are as shown:
SwimmerMeanSD1 backstroke50.720.242 breaststroke55.210.223 butterfly48.430.254 freestyle44.910.21
a) What are the mean and standard deviation for the relays total time in this event?
b) The Olympic record so far was 3:19.48. (Thats 199.48 seconds) Do you think the team is likely to swim faster than this at the Olympics? Show your work and explain.
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