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Probability is about measuring the likelihood of an event happening. Imagine flipping a coin; getting heads is one possible outcome. Probability assigns a number between 0 and 1 to events to show how likely they are to happen. A probability of 0 means an event is impossible, while a probability of 1 means it's certain. For instance, in the context of the exercise, the probability of a graduate being a business major is calculated as follows: among 30 graduates, 11 are business majors, so the probability is \(P(\text{B}) = \frac{11}{30}\).

Think of probability as a way to predict outcomes over the long run. In practical terms, if hundreds or thousands of students were hired, we would expect about one-third of them to be business majors, given the probability here. Remember that each event or outcome is unique, but the patterns emerge as you look at large numbers of trials or cases.

Conditional probability considers how the probability of one event is affected by the occurrence of another event. It's about answers colored by extra context or information. Let's say you want to know the probability of pulling a red card out of a deck, given that you've already drawn one.

Similarly, in the exercise, we wanted to know the probability of a graduate being a business major, given that they are female. This is written as \(P(\text{B|F})\). Out of the 16 female graduates, 9 are business majors, so \(P(\text{B|F})= \frac{9}{16}\).

• The condition limits the total pool of possibilities.
• It's called 'conditional' because calculations depend on additional criteria, such as being a female in this case.

Conditional probability highlights how event outcomes might depend on particular circumstances.

Events in probability can be independent, dependent, or mutually exclusive. Understanding these relationships helps determine how two or more events might influence one another or how they simply cannot occur simultaneously.

**Independent events**: Two events are independent if the occurrence of one does not affect the probability of the other. Earlier, calculating \(P(\text{B})\) and \(P(\text{B|F})\) showed they are not equal, indicating the events "female" and "business major" are not independent.

**Mutually exclusive events**: These events cannot occur together—if one happens, the other cannot. In our exercise, a person can be both a female and a business major, so these events aren't mutually exclusive.

• When assessing relationships, calculate probabilities to see if conditions affect outcomes.
• Think about whether one event happening prevents another. Do they need to be examined together or separately?

Evaluating events in probability is like troubleshooting—you're testing if certain conditions lead to specific outcomes, or if two events are linked together or distinctive.