概率、独立性与条件概率

Sample spaces and events

Adding probabilities the right way

Conditioning shrinks the sample space

Independence and its traps

Roll two dice. Use the complement rule to find the probability that at least one die shows a 4, and identify the complement event.

An urn holds six blue and four red balls. Two balls are drawn without replacement. What is the probability both are blue, and which computation produces it?

Roll a four-sided die, then flip that number of coins. Using the partition formula with the four possible rolls, the probability of exactly one heads works out to which value?

Let A be the event the first die shows 4 when two dice are rolled. Which of these events is independent of A?

Three coins are flipped. Let A be the first two match, B the last two match, and C the first and last match. Which statement about A, B and C is true?

From the urn with six blue and four red balls, the first drawn ball is blue. What is the probability the second draw is also blue, and why?

Roll two dice. Compute the probability that neither die shows a 4, that is the complement of seeing at least one 4, and write the exact fraction.

From the urn with six blue and four red balls, given that the first ball drawn is blue, compute the conditional probability that the second ball drawn is blue, and write the exact fraction.

Three fair coins are flipped. Compute the probability that the first two match and the last two match and the first and last match, the triple intersection of the lesson's three events, and write the answer.