Final answer:
The statement 'If two numbers are odd, then their sum is even' is true. For the die roll scenarios, the probability and relationship between events require analyzing the outcomes of each event and their intersections to determine if they are mutually exclusive or independent.
Step-by-step explanation:
Regarding the initial student question, if two numbers are odd, then their sum is even, the statement is true. This can be demonstrated using the properties of integers. Odd numbers can be expressed in the form of 2n+1, where n is an integer. When you add two odd numbers, you basically add (2n+1) + (2m+1), where n and m are integers, and the result is 2n + 2m + 2, which simplifies to 2(n + m + 1). Since n + m + 1 is an integer, the sum is a multiple of 2, hence even.
For the roll of a die exercises, considering event A as rolling either a three or four first, followed by an even number, to find P(A) you'd consider the probability of the first roll (1/6 for a three, 1/6 for a four) and the probability of the second roll being even (1/2), making the total probability P(A) = (1/6 + 1/6) * (1/2). The event B, where the sum of two rolls is at most seven, involves counting all the possible combinations of rolls that amount to seven or less and then dividing by the total number of outcomes, which is 36 (6 faces on the first die times 6 faces on the second die). To find the probability P(A|B), representing the probability of event A given B has occurred, you'd need to calculate the number of favorable outcomes for A that also satisfy B, and divide by the number of outcomes for B alone.
Based on the descriptions of events A and B, we can say that these events are not mutually exclusive, as there are outcomes in which both can occur, such as rolling a three followed by a four, which equals seven. To determine if events A and B are independent events, one must assess if the occurrence of one affects the probability of the occurrence of the other. If the probability of A happening is the same whether B has occurred or not, then they are independent; otherwise, they are not.