(a) An urn contains four black and five white balls. A ball...

(a) An urn contains four black and five white balls. A ball is drawn at random and then replaced,after which a second ball is drawn. Find the probability that the first is black and the second is white. (b) If a pair of fair dice is rolled, what is the probability that the sum shown is either exactly 9 or less than 4 ? (c) A coin is tossed, then a card is drawn from an ordinary deck of 52 eards, and then a die is rolled. What is the probability of obtaining a head on the coin, an ace from the deck and a 5 on the die? (d) If one letter is chosen at random from the word boot and one letter from the word foot, what is the probability that both letters match?

(a) An urn contains four black and five white balls. A ball is drawn at random and then replaced,after which a second ball is drawn. Find the probability that the first is black and the second is white.
(b) If a pair of fair dice is rolled, what is the probability that the sum shown is either exactly 9 or less than 4 ?
(c) A coin is tossed, then a card is drawn from an ordinary deck of 52 eards, and then a die is rolled. What is the probability of obtaining a head on the coin, an ace from the deck and a 5 on the die?
(d) If one letter is chosen at random from the word boot and one letter from the word foot, what is the probability that both letters match?

Key Concepts

Replacement in Probability Experiments

Replacement is a methodological concept in probability that involves returning an item to the sample space after it is selected. This practice ensures that the conditions remain unchanged for subsequent selections, preserving the independence of events and stabilizing the probability calculations across multiple trials.

Sample Space and Counting Techniques

The sample space encompasses all possible outcomes of a random experiment, and counting techniques help determine both the total number of outcomes and the number of favorable outcomes. Mastery of these techniques is crucial for accurately calculating probabilities in various scenarios.

Addition Rule for Mutually Exclusive Events

The addition rule is applied when events are mutually exclusive, meaning they cannot occur simultaneously. In such cases, the probability that one or another occurs is the sum of their individual probabilities. This rule simplifies the process of calculating probabilities for non-overlapping outcomes.

Multiplication Rule for Independent Events

The multiplication rule states that if events are independent, the probability of all the events occurring is the product of their probabilities. This rule is useful in calculating the likelihood of compound events where multiple independent actions or experiments take place consecutively.

Independent Events

Independent events are occurrences where the outcome of one event does not affect the outcome of another. This concept is vital in probability theory because it allows the overall probability of multiple events happening consecutively to be computed by multiplying their individual probabilities.

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