Final answer:
Probability measures the likelihood of an event occurring. The probability of the symmetric difference concerns the chance of exactly one event occurring, which is different from mutual exclusivity, where events can't occur together, and independence, where one event's occurrence doesn't affect another's probability.
Step-by-step explanation:
The concept of probability is the measure of how likely an event is to occur. Specifically, when we discuss the probability of the symmetric difference, we're looking at the probability of exactly one event occurring out of two unique events.
For example, in problem b, event A is rolling a three or four first, followed by an even number. To find P(A), you must calculate the possibility of these combined outcomes occurring during the rolls. Whereas, event B is having a sum of two rolls that is at most seven. To find P(B), you evaluate the likelihood of this outcome.
The term P(A|B) represents the probability of event A occurring given that event B has already occurred, known as conditional probability.
Two events are mutually exclusive if they cannot occur at the same time, such as in examples where event A is choosing a blue card and tossing a head, and event B is choosing a red or green card and tossing a head. If events A and C involve choosing a blue card and a red card respectively, and both involve tossing a head, they are not mutually exclusive since the events do not prevent each other from occurring.
Independent events like A and B in an experiment implies that the occurrence of one event does not affect the probability of the other occurring. Their probability does not influence each other.