Final answer:
The probability of each outcome on a six-sided number cube is ⅓. Event A's probability P(A) is ⅓, and event B involves combinations for a sum at most seven. P(A|B) represents the probability of A given B, and A and B are neither mutually exclusive nor independent.
Step-by-step explanation:
When rolling a six-sided number cube, the probability of each individual outcome (such as getting a 3) is ⅓ since there is one favorable outcome out of six possible outcomes. Therefore, P(3) is ⅓. To find P(1,2, or 3), which is the probability of rolling a 1, a 2, or a 3, we add the individual probabilities since these are mutually exclusive events: P(1,2, or 3) = ⅓ + ⅓ + ⅓ = ½. Similarly, P(less than 4) is also ½ since it includes the same outcomes as P(1,2, or 3).
For question b, event A requires rolling a 3 or a 4 first (2 outcomes), followed by an even number (3 outcomes: 2, 4, 6). So, P(A) = (⅓) × (⅓) = ⅓. In question c, event B refers to the sum of two rolls being at most seven. This includes multiple combinations, and with each die having a ⅓ chance for any number, the computations involve counting all the pairs of outcomes that yield a sum of seven or less.
To understand P(A|B), it represents the probability of event A happening given that event B has happened. This conditional probability is calculated by considering the overlap between A and B's favorable outcomes divided by the total number of outcomes in B.
Events A and B are not mutually exclusive because it's possible for both to occur simultaneously, for example, rolling a 3 followed by a 4. As for independence, if knowing whether event B has occurred does not change the probability of A occurring (and vice versa), the events are independent. Numerical justification will require considering the definition of independent events: P(A and B) = P(A) × P(B).