You have just won 64,000on the TV quiz show "who wants to be a millionaire? You can either quit with those winnings or risk answering another question if your answer is correct then you win 128,000, if incorrect your winnings fall to 32,000. Using the expected value of winnings and taking only these three values into account question if and only if I am at least__________percent confident of answering it correctly

The research division of pharmaceutical company has already spent 4000,000 to determine whether a new medication for the prevention of seasickness is effective. Now the director of the division must decide whether to spend an additional 200,000 to complete the test, knowing that the probability of success is only 1/3. He also knows that if the tests are continued and the medication proves to be effective, this would result in new revenues of 2,100,000, for a net profit of 1,500,000 to his company. Of course, if the tests are continued and the medication turns out to be ineffective, this would mean 600,000 “down the drain”. He also knows that if the tests are not continued and the medication is successfully produced by a competitor, this would entail an additional loss of 100,000 due to his company's expected profit?


Two equally matched teams play a best –of-7 series. What is the expected length of the series?

A fair die is tossed repeatedly, and each outcome is recorded. What is the expected number of tosses before the first repeat occurs? (“Repeat” means something that has already been tossed)

During world war 2, on any given bombing run over Germany by U.S. B-17 plane, the probability of the plane being shot down was 4.5%
A) What would be the probability of a crewman surviving 25 such bombing runs, the stipulated tour of duty?
B) what tour of duty would have given a survival rate closest to 50%
C) What would be the life expectancy of a crewman, in average number of bombing run?

According to the State Lobster Hatchery of Massachusetts, approximately 1 out of every 20,000,000 lobster hatched is blue.
a) How many lobsters must be hatched before we can be at least 90% confident that at least 1 is blue?
b) If 100,000,000 lobsters were hatched, what would be the probability that between 10 and 20 were blue?

The standard American roulette wheel consist of 38 slots, numbered 0, 00 and 1 through 36. The slots numbered 0 and 00 are green, of the remaining slots, exactly half are Red and half are black. On each ply, a ball is bounced on the turning wheel and it land randomly in one of the slots. Players are allowed to bet on specific numbers, specific colors, or in other words. Suppose that you walk into a casino and repeatedly play the roulette wheel, each time betting on either Red or black. Your rule is to quit if and only if you suffer a string of 6 consecutive losses. What is the probability of quitting before your 93rd play?

The Art of Shakespeare's sonnets (Harvard Univ. press 1997) author Helen Vendler noted that each of the 14 lines of sonnet 20 (one of 154 sonnet written by Shakespeare) includes the letter of the word “hues” and/or the letters of the world “hews”.
Suppose that 154 moneys sitting at 154 keyboards pounded out one sonnet apiece, each sonnet consisting of 14 lines of 36 alphabet letters each with each letter equally likely. What is the probability that in at least one of the sonnets, every line includes the letters of the word “hues” and/or the letters of the word “hews”?

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