(a) There are 3 red and 5 black balls in one box and 6 red...

(a) There are 3 red and 5 black balls in one box and 6 red and $+$ white balls in another. If you pick a box at random, and then pick a ball from it at random, what is the probability that it is red? Black? White? That it is either red or white? (b) Suppose the first ball selected is red and is not replaced before a second ball is drawn. What is the probability that the second ball is red aiso? (c) If both balls are red, what is the probability that they both came from the same box?

(a) There are 3 red and 5 black balls in one box and 6 red and $+$ white balls in another. If you pick a box at random, and then pick a ball from it at random, what is the probability that it is red? Black? White? That it is either red or white?
(b) Suppose the first ball selected is red and is not replaced before a second ball is drawn. What is the probability that the second ball is red aiso?
(c) If both balls are red, what is the probability that they both came from the same box?

Key Concepts

Law of Total Probability

This concept involves breaking down the calculation of probabilities into different disjoint cases that cover all possible scenarios. In problems where a choice must be made among several groups or conditions (such as choosing one of multiple boxes), the overall probability is the sum of the probabilities for each case, weighted by the probability of each case occurring.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is especially pertinent in multi-stage experiments or sequential draws where the outcome of one event affects the probabilities in subsequent events.

Compound Events

Compound events involve the outcomes of two or more separate events occurring together. In probability problems that require determining outcomes such as selecting a ball from one box and then another from the same or a different box, the overall probability is determined by multiplying probabilities along the stages, with adjustment for any dependence between events.

Without Replacement

When sampling without replacement, objects are not returned to the set after being drawn, which changes the composition of the set and thus the probabilities for subsequent events. This concept is important for understanding how the odds of drawing a particular item change based on previous outcomes in sequential draws.

Bayes' Theorem

Bayes' Theorem is used to update the probability estimate for a hypothesis as additional evidence is acquired. In problems where you are required to determine the reverse conditional probability (for example, the probability that a specific box was chosen given the outcome of the draws), Bayes' Theorem provides the necessary framework to reverse conditional probabilities.

Please give Ace some feedback