Quiz: Probability

Is the following statement True or False?

If P(a|b,c) = P(b|a,c), then P(a|c) = P(b|c)

Is the following statement True or False?

P(a|b∧a) = 1

Is the following statement True or False?

If P(a|b,c) = P(a), then P(b|c) = P(b)

Is the following statement True or False?

P(a|b) = P(a), then P(a|b,c) = P(a|c)

Suppose you are a witness to a nighttime hit-and-run accident involving a taxi in Athens. All taxis in Athens are blue or green. You swear, under oath, that the taxi was blue. Extensive testing shows that, under the dim lighting conditions, discrimination between blue and green is 75% reliable.

It is possible to calculate the most likely color for the taxi given the current information. (Hint: distinguish carefully between the proposition that the taxi is blue and the proposition that it appears blue.)

Suppose you are a witness to a nighttime hit-and-run accident involving a taxi in Athens. All taxis in Athens are blue or green. You swear, under oath, that the taxi was blue. Extensive testing shows that, under the dim lighting conditions, discrimination between blue and green is 75% reliable.

If you know that 9 out of 10 Athenian taxis are green, what is the most likely color for the taxi?

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Above is a table listing the probabilities of three binary random variables. For the table cell marked with A, fill in the correct value. Your answers will be evaluated to 4 decimal places. Use a period as a decimal separator.

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Above is a table listing the probabilities of three binary random variables. For the table cell marked with B, fill in the correct value. Your answers will be evaluated to 4 decimal places. Use a period as a decimal separator.

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Above is a table listing the probabilities of three binary random variables. For the table cell marked with C, fill in the correct value. Your answers will be evaluated to 4 decimal places. Use a period as a decimal separator.

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Above is a table listing the probabilities of three binary random variables. For the table cell marked with D, fill in the correct value. Your answers will be evaluated to 4 decimal places. Use a period as a decimal separator.

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Above is a table listing the probabilities of three binary random variables. For the table cell marked with E, fill in the correct value. Your answers will be evaluated to 4 decimal places. Use a period as a decimal separator.

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Above is a table listing the probabilities of three binary random variables. For the table cell marked with F, fill in the correct value. Your answers will be evaluated to 4 decimal places. Use a period as a decimal separator.

Suppose you are given a bag containing n unbiased coins. You are told that n−1 of these coins are normal, with heads on one side and tails on the other, whereas one coin is a fake, with heads on both sides.

Suppose you reach into the bag, pick out a coin at random, flip it, and get a heads. What is the (conditional) probability that the coin you chose is the fake coin?

Suppose you are given a bag containing n unbiased coins. You are told that n−1 of these coins are normal, with heads on one side and tails on the other, whereas one coin is a fake, with heads on both sides. Suppose you reach into the bag, pick out a coin at random, flip it, and get a heads. Suppose you continue flipping the coin for a total of k times after picking it and see k heads. Now what is the conditional probability that you picked the fake coin? Suppose k=n=12, round to 4 decimals. Use a period as a decimal separator.

You are given three statements.

1. P(X,Y|Z) = P(X|Z)P(Y|Z)

2. P(X|Y,Z) = P(X|Z)

3. P(Y|X,Z) = P(Y|Z)

True or False: statement 1 is equivalent to (statement 2 ∧ statement 3).

Consider two medical tests, A and B, for a virus. Test A is 95% effective at recognizing the virus when it is present, but has a 10% false positive rate (indicating that the virus is present, when it is not). Test B is 90% effective at recognizing the virus, but has a 5% false positive rate. The two tests use independent methods of identifying the virus. The virus is carried by 1% of all people. Say that a person is tested for the virus using only one of the tests, and that test comes back positive for carrying the virus. Which test returning positive is more indicative of someone really carrying the virus?
After your yearly checkup, the doctor has bad news and good news. The bad news is that you tested positive for a serious disease and that the test is 99% accurate (i.e., the probability of testing positive when you do have the disease is 0.99, as is the probability of testing negative when you don’t have the disease). The good news is that this is a rare disease, striking only 1 in 10,000 people of your age. What are the chances that you actually have the disease? Give your answer as a value between 0 and 1, and round your answer to 6 decimals. Use a period as a decimal separator.
Two factories - Factory A and Factory B - design batteries to be used in mobile phones. Factory A produces 60% of all batteries, and Factory B produces the other 40%. 2% of Factory A’s batteries have defects, and 4% of Factory B’s batteries have defects. What is the probability that a battery is both made by Factory A and defective? Round your answer to 3 decimal digits. Use a period as a decimal separator.
Consider a standard 52-card deck of cards with 13 card values (Ace, King, Queen, Jack, and 2-10) in each of the four suits (clubs, diamonds, hearts, spades). If a card is drawn at random, what is the probability that it is either a spade or a two? (Note that “or” in this question refers to inclusive, not exclusive, or). Round your answer to 3 decimal digits. Use a period as a decimal separator.
Imagine flipping two fair coins, where each coin has a Heads side and a Tails side, with Heads coming up 50% of the time and Tails coming up 50% of the time. What is probability that after flipping those two coins, one of them lands heads and the other lands tails? Round your answer to 3 decimal digits. Use a period as a decimal separator.
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