Probability problem3
Swagman888
probability_questions.docxThere are five problems in this assignment for which you should submit solutions.
Please clearly label all problems and show all work! In particular, some questions specifically call for explanations. These are important.
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Problems to submit:
1. Pat and Sam are arguing about the probability of obtaining 2 heads and 2 tails in 4 flips of a fair coin, in that specific order. Pat says, "The probability equals 1/16, since there is one way this outcome can occur, and the experiment has 16 different possible outcomes". Sam says "The probability equals 6/16, since the corresponding entry in row 4 of Pascal's triangle is 6, and the sum of the entries in that row equals 16." Who is correct, and why?
2. Explain briefly, in your own words, why the number of outcomes in which we obtain 5 heads and 5 tails in 10 flips is equal to the following:
the number of outcomes in which we obtain exactly 4 heads in 9 flips, plus the number of outcomes in which we obtain 5 heads in 9 flips.
3. Suppose that we aren’t sure whether a certain coin is actually fair or not, and to test it for fairness, we apply the following procedure: Flip the coin a certain number of times, and
· if the number of heads and the number of tails differ from one another by at most 2, we conclude that the coin is fair;
· if the number of heads and the number of tails differ from one another by more than 4, we conclude that the coin is not fair;
· otherwise the test is inconclusive.
a. If the coin really is fair, and we flip 10 times during the test, what is the probability of drawing the correct conclusion from the test, that the coin is fair?
b. If the coin really is fair, what is the probability of incorrectly concluding (based on 10 flips) that the coin is not fair?
c. If the coin really is fair, does flipping 12 times instead of 10 during the test increase the probability of drawing the correct conclusion, decrease this probability, or leave it unchanged?
4. According to a simplified model, a certain stock goes either up $1 (U) or down $1 (D) each day, and is equally likely to go up or down each day. Suppose that we observe the sequence of ups and downs of this stock for 9 days.
a. What is the probability that at the end 9 days, the stock price will have decreased by exactly $5?
b. What is the probability that at the end of 9 days, the stock price will have had a net change of at most $3 (in either direction)?
5. You have three options in a certain game:
Option A: Flip 10 fair coins. If there are 4, 5, or 6 heads, then you receive $100. Otherwise you lose $150.
Option B: The exact opposite of option A. Flip 10 fair coins. If there are 4, 5, or 6 heads, then you lose $100. Otherwise you receive $150.
Option C: Do nothing.
Of these options, which one, if any, has a positive expected gain, and how much is that expected gain?
