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EL6303 (Elza Erkip) HW 1 Fall 2015 1. For any arbitrary events , ,A B C with ( ) 0P C > , prove or disprove that

( )(( )| ) 1 ( )P A B CP A B C P C≥ − U UU .

(Video is required.)

2. Given: 0 ( ) 1, 0 ( ) 1.P A P B< < < < Prove ( | ) ( | ) 1P A B P A B+ = iff A, B are independent. (Video is required.) 3. A digital signal “1” or “0” is transmitted through a noisy channel, the

received data may be different from the signal sent out. Suppose the transmitter sends out “0” with probability 0.6, and “1” with probability 0.4. When “0” is transmitted, the receiver receives “0” with probability 0.7, and “1” with probability 0.3. When “1” is transmitted, the receiver receives “0” with probability 0.2, and “1” with probability 0.8.

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(1) Find the probability that data “0” is received. (2) Find the probability of error (i.e. the probability that “1” is received

when “0” was transmitted or “0” is received when “1” was transmitted).

4. Given a right triangle with sides a=3, b=4, c=5. Draw a circle with radius R, tangent to sides a and c, as in the figure below, where R is a random variable uniform in (0, 3).

Find the probability that the circle stays inside of the triangle.

5. Suppose 100 people are waiting for a blood test for a kind of disease. The

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probability that for one person the test result is positive equals 0.1. Two test methods are as follows: Method 1: Test each person one by one. Then we have to do 100 tests.

Method 2: Divide 100 people into 10 groups with 10 people in each group. Then, mix 10 people’s blood and test that mixture. If the result is negative, then everyone in this group is negative. If the result is positive, then we test these 10 people’s blood one by one. Now, the number of tests becomes 1+10=11. Question: On the average, what is the number of the tests we need to do by the second method?

6. In the woods, a hunter is shooting at a hare. The probability of success for

his first shot is ½. If he misses the first shot, the probability of success for his second shot is 1/8. If he misses again, the hare runs away. What is the probability that he will hit the hare?

7. The events A and B are mutually exclusive and P(A)≠0, P(B)≠0. Can they

be independent? Prove your answer.

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8. A train and a bus arrive at the station at random between 9 AM and 10 AM. The train stops for 5 minutes and the bus for x minutes. Find x so that the probability that the bus and the train will meet equals 0.2.