- DATA.ML.310
- 5. Probability and Probabilistic Inference
- 5.2 Quiz: Probability

# 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?

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.

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.

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.

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.

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.

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?

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).