Probability

Why we can multiply: For independent events, knowing that one happens does not change the chance of the other. By definition, this gives the multiplication rule Substitute the given probabilities: So the probability that both and occur together is .

Why use conditional formula: By definition, Substitute values: So, given that has occurred, the chance that also occurs is .

Reason. Use the addition law: Compute:

Step 1: Use the formula for conditional probability: Step 2: Substitute the given values: Final Answer:

Step 1: This is a binomial probability problem. The probability of getting exactly heads in tosses is given by: Step 2: For a fair coin: - - - Step 3: Substitute values: Final Answer:

Step 1: Recall the formula relating union and intersection of two events: Step 2: Rearranging for : Step 3: Substitute the given values: Final Answer:

Step 1: By definition, two events and are independent if and only if: Step 2: This means the occurrence of one event does not affect the probability of the other. Step 3: Let's briefly check the other options: - (B) is always true for any two events (not just independent). - (C) implies both events are impossible, which is not implied by independence. - (D) is incorrect unless and are mutually exclusive and at least one of them has probability 0. Final Answer:

Step 1: In a standard deck of 52 playing cards, there are 4 kings (one from each suit: hearts, diamonds, clubs, and spades). Step 2: Probability is calculated as: Final Answer:

The mean is given by:
Final answer:

A doublet occurs when both dice show the same number. The probability of getting a doublet in a single throw is: The probability of not getting a doublet in a single throw is: We are asked to find the probability of getting at least two doublets in four throws. This is a binomial probability problem, where: - (the number of trials), - (the probability of success, i.e., getting a doublet), - (the probability of failure, i.e., not getting a doublet). The probability of getting exactly doublets in 4 throws is given by the binomial distribution: We need to calculate the probability of getting at least two doublets, i.e., , which can be expressed as: Now, calculate and : 1. For (no doublets): 2. For (one doublet): Now, calculate : Therefore, the probability of getting at least two doublets in four throws is:

Given: and Recall: Now use the formula: Convert to common denominator 26: % Final Answer Answer:

The addition theorem of probability states that the probability of the union of two events and is given by: This formula accounts for the fact that the overlap (intersection) of and is counted twice when we simply add and , so we subtract to correct for that.

The odds in favor of an event occurring are given as , meaning that for every favorable outcomes, there are unfavorable outcomes. The probability of event is given by: Thus, the correct answer is option (B) .

The multiplication theorem of probability, which is used to find the probability of the intersection of two events and , states that: This formula is derived from the definition of conditional probability. The conditional probability is the probability of event occurring given that event has already occurred. Therefore, the joint probability of and happening is the product of the probability of and the conditional probability of given .

A doublet is the event in which both dice show the same number. The possible outcomes when two dice are thrown are: The favorable outcomes for a doublet are: Thus, there are 6 favorable outcomes. Therefore, the probability of getting a doublet is: Thus, the correct answer is option (B) .