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Homework 2: Math 493

All of the questions are worth equal points. The homework is due on Sep 27th.

1. Conditional Probability Problems

• (Three Prisoners Problem) Three prisoners, A, B, and C, are on death row. The governor decides to pardon one of three and chooses at random the prisoner to pardon. He informs the warden of his choice but requests that the name be kept secret for a few days. The next day, A tries to get the warden of his choice but requests that the name be kept secret for a few days. The next days, A tries to get the warden to tell him who had been pardoned. The warden refuses. A then asks which of B or C will be executed. The warden thinks for a while, then tells A that B is to be executed. Warden thinks that he has no given extra information to A about him getting pardoned. Compute the conditional probability that A has been pardoned given that the Warden says that B will be executed. Is the Warden’s reasoning sound?

• (Urn Problem) Balls are randomly removed from an urn that initially contains 20 red balls and 10 blue balls. What is the probability that all of the red balls are removed before all of the blue ones have been removed?

2. Bayes’ Rule

• A doctor is called to see a sick child. The doctor has prior information that 90% of sick children have the flu, while the other 10% are sick with measles. A well-known symptom of measles is a rash R. Assume that the probability of having a rash if one has measles is 0.95. However, occasionally children with flu also develop rash, and the probability of having a rash if one have flu is 10%. Upon examining the child, the doctor finds a rash. What is the probability that the child has measles?

• Suppose there are A, B, C medical testing kits that are available for testing the presence of Covid-19 strain. Suppose the false positive probability of detecting the strain for these testing kits are 5%, 10% and 20%, i.e., incorrectly detecting the presence of Covid-19 strain when the underlying person is healthy. Assume the prevalent infection rate of Covid-19 strain to be 1%. Compute the probability that the person is healthy given that one of the three testing kit is positive.

• Show that

Suppose that before new evidence is observed, the hypothesis is three times as likely to be true as is the hypothesis. If the new evidence is twice as likely when G is true than it is when H is true, which hypothesis is more likely after the evidence has been observed.?

3. Random Variable

• Suppose a coin is tossed twice so that the sample space is S={HH, HT, T H, T T}. Assume that you are interested in counting the number of heads. Can you define a random variable X corresponding to the number of tails in the sample space.?

• In the previous question, suppose you were supposed to toss the coin as long as you do not get one tail. However, if were defining the random variable Y as the number of head then what is your random variable Y ?

• Suppose instead of tossing two coins, you toss a coin, a die, and draw a card from a deck of cards. Assume that the each of the three draws are pair-wise independent. Then, what is the probability that you will observe a {H, 6, Ace}.

4. Cumulative Distribution Function

• The distribution function of the random variable X is given by

Draw the graph of the above C.D.F. Also, compute i) P(1 < X < 3) and P(1 ≤ x < 3). Are they different, if so, why?

• (Geometric Distribution) Suppose we do an experiment that consists of tossing a coin until a head appears. Let p = probability of a head on any given toss, and define a random variable X = number of tosses required to get a head. Then, for any x = 1, 2, · · · ,

P(X = x) = (1 − p) x−1 p,

since we must get x − 1 tails followed by a head for the event to occur and all trials are independent. Derive the cumulative distribution function for X?

5. Expected Values

• For the geometric distribution question, could you compute the expected value of X? Could you interpret it?