Need help with some Probabilistic Reasoning questions any helps works! 1. Probab

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Need help with some Probabilistic Reasoning questions any helps works!

1. Probability I (10 points) After your yearly checkup, the doctor has good news and bad news. The bad news is that you tested positive for a serious disease and that the test is 99% accurate (ie, the probability of testing positive when you do have the disease is 0.99, as is the probability of testing negative when you do not have the disease). The good news is that this is a rare disease, striking only 1 in 10,000 people of your age. What is the probability that you actually have the disease? Show your work. 2. Probability II (12 points) A space probe near Neptune communicates with Earth using bit strings. Suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. When a 0 is sent, the probability it is received correctly is 0.9, and the probability it is received incorrectly (as a 1) is 0.1. When a 1 is sent, the probability that it is received correctly is 0.8, and the probability it is received incorrectly (as a 0) is 0.2. (a) What is the probability that a 0 is received? Show your work. (b) What is the probability that a O was transmitted given that a 0 was received? Show your work. 3. Probability III (12 points) Suppose we have two random variables, both defined over the students in CS 4365 wh worked hard for the course . ga got an A

Explanation / Answer

1.

Consider a sample of 10,000 people. Of them, 1 person has the disease. If 10,000 people take the test, there is a 1% chance of testing positive if you DON'T have the disease. So, 100 people will test positive.


P (test|disease) = 0.99
P (¬test|¬disease) = 0.99
P (disease) = 0.0001

P (disease|test) = P (test|disease)P (disease) /
( P (test|disease)P (disease) + P (test|¬disease)P (¬disease))

= 0.99 × 0.0001 / (0.99 × 0.0001 + 0.01 × 0.9999)
= .009804

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