Question

Suppose that marvin rolls a pair of fair six-sided dice. Let \[a\] be the event that the first die lands on \[4\] and \[b\] be the event that marvin rolls doubles. Using the sample space of possible outcomes below, answer each of the following questions. What is \[p(a)\], the probability that the first die lands on \[4\]? what is \[p(b)\], the probability that marvin rolls doubles? What is \[p(a\text{ and }b)\], the probability that the first die lands on \[4\] and marvin rolls doubles? are events \[a\] and \[b\] independent? choose 1 answer: choose 1 answer: (choice a) yes, events \[a\] and \[b\] are independent events. a yes, events \[a\] and \[b\] are independent events. (choice b) no, events \[a\] and \[b\] are not independent events. b no, events \[a\] and \[b\] are not independent events.

Answer 1

In probability theory, independent events do not affect each other's likelihood of occurring. Event A pertains to rolling either a three or four first, followed by an even number, while Event B involves rolling a sum of at most seven. Numerical justification is needed to determine if A and B are either mutually exclusive or independent events in this scenario.

Step-by-step explanation:

In mathematics, particularly in probability theory, events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. This can be mathematically expressed as P(A ∩ B) = P(A) × P(B). To determine the independence of two events, one must ascertain if this equation holds true.

Regarding the dice example, event A is rolling either a three or four first, followed by an even number. To calculate P(A), we take into consideration the event 'rolling a 3 or 4' which has a probability of 2/6 (since there are two favorable outcomes out of six total outcomes for one die). The probability of rolling an even number is 1/2 because there are three evens out of six possible outcomes. Because these two events are independent (rolling a number on the first die does not affect the outcome of the second), we multiply their probabilities: P(A) = (2/6) × (1/2) = (1/3) × (1/2) = (1/6).

Event B is the event where the sum of the two dice is at most seven. To find P(B), we recognize that there are several combinations of numbers that satisfy this condition. One could calculate this by enumerating all possible outcomes that result in a sum of 7 or less (e.g., (1,1), (1,2), etc.), and dividing this by the total number of outcomes (36). Importantly, some outcomes are favorable for both A and B, which means they are not mutually exclusive.

P(A|B) represents the probability that event A occurs given that B has already occurred. This can be calculated as P(A ∩ B) / P(B). We would first identify the outcomes that satisfy both A and B, and then compute the ratio of their number to the number of outcomes that satisfy B alone.

Events A and B are mutually exclusive if they can never occur simultaneously. Numerical justification involves demonstrating the presence or absence of common outcomes between A and B. Similarly, to determine if events A and B are independent, we would check if P(A|B) = P(A). If this equation holds, A and B are independent; otherwise, they are not.