probabiltiy

 

 only questions 1,2,4(a), 5, 7 are needed to be done. 

 

 typed answer either in doc or in pdf. 

  

 strict due. by EST 3am (09/12)

Math 464 – Fall

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3 – Homework

2

1. In the 1990’s the ELISA test was used to test blood donations for the
AIDS virus. If a sample of blood has the virus, the test will be positive
99.9% of the time. If the sample does not have the virus the test will be
negative 99.5% of the time. Suppose that 0.2% of all the samples of donated
blood have the virus. We pick a sample and when we test it, the test is
positive. What is the probability that the sample has the virus?

2. Four mathematicians are hiding a coin in each of their hands. The coins
are all silver or gold. Two of them have one gold and one silver coin. One
of them has two gold coins. One of them has two silver coins. We pick a
mathematician at random and pick one of his or her hands at random. The
coin in that hand is revealed to be gold.
(a) What is the probability the coin in the other hand is also gold?
(b) What is the probability the coin in the other hand is silver?

3. This example shows that pairwise independence does not imply indepen-
dence. It is problem 4 from chap 1 of the book. We roll a fair six-sided die
twice. Let A be the event that the first roll is odd, B the event that the sec-
ond roll is even, and C the event that either both rolls are even or both rolls
are odd. Show that A, B, C are pairwise independent but not independent.

4. (Exposition) Do one of the following problems. The second one is more
challenging.
(a) We flip a fair coin n times. Let u

n
be the probability that there is no run

of 4 heads (i.e., 4 heads in a row) in the n flips. Find a recursion relation
that gives u

n
in terms of u

n−1, un−2, un−3 and un−4. Use it to compute u8.

(b) A biased coin has probability p of heads. (So the probability of tails is
1 − p.) We toss it until we get a run of r heads or a run of s tails. Let E be
the event that the run of r heads occurs before the run of s tails. Our goal
is to compute P(E). Let A be the outome of the first toss. Show

P(E|A = heads) = pr−1 + (1 − pr−1)P(E|A = tails)

Find a similar expression for P(E|A = tails) and use your two equations to
find P(E).

5. A random variable has a Poisson distribution with parameter λ. (This is
defined in section 2.2 of the notes.) Compute the following probabilities.
(a) P(2 ≤ X ≤ 4)

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(b) P(X ≥ 4)
(c) P(X is even)

In each case you should give an exact answer and a decimal approximation
to your answer when λ = 2 that is correct to 3 decimal places.

6. There is an R program in the file “hmwk2 r.txt” that you can find on
the homework page. It computes an approximate answer to part (c) of prob-
lem 5. It uses the R function rpois() to generate samples of the Poisson
random variable. In R, %% is the “mod” operation. (n%%2)==0 is true
if and only if n is even. You can run the program from within R by the
command source(“hmwk2 r.txt”,echo=TRUE). Modify the program so that
it computes approximately the answers to parts (a) and (b) of problem 5.
Your solution should include a copy of the program and the output you got
when you ran it. If you want to do this from scratch in another software
package that is fine. Include a copy of the program and be sure to say what
package you are using.

7. (Exposition) Prove that if A and B are independent events, then
(a) A and Bc are independent.
(b) Ac and B are independent.
(c) Ac and Bc are independent.

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