Final answer:
To derive the equation for consecutive even integers, define the first integer as 2n and add 2 for each subsequent integer, resulting in a sequence: 2n, 2n + 2, 2n + 4, etc. In probabilities, when seeking the likelihood of combined events, list the outcomes that meet all conditions and compute both individual and joint probabilities.
Step-by-step explanation:
Deriving the equation for consecutive even integers starts with recognizing that even integers are multiples of 2. If n represents any integer, then 2n is an even integer. For consecutive even integers, you simply add 2 to the previous even integer. Thus, if 2n is our first even integer, then the next consecutive even integer is 2n + 2, the one after that is 2n + 4, and so on. This pattern can be represented as the sequence: 2n, 2n + 2, 2n + 4, ... where n is an integer.
Focusing on the probability aspect, if a sample space S has only six outcomes, and you are looking for the number of outcomes that are both 2 or 3 and even, you must look at the common elements that satisfy both conditions. If S consists of whole numbers starting at one and less than 20, to find event A (the even numbers in S) and event B (numbers greater than 13 in S), you would list the elements that fulfill these criteria. P(A) would denote the probability of event A, while P(A AND B) would denote the joint probability of both events occurring.