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Probability is a way to measure the likelihood of an event occurring. It helps us understand how certain or uncertain things are. For any given event, probability values range from 0 (impossible) to 1 (certain).
For example, when flipping a coin, the probability of landing on heads is 0.5, as there are two equally likely outcomes: heads and tails.
In general terms, the probability of an event is calculated as:

• \( P(A) = \frac{\text{Number of favorable outcomes for event A}}{\text{Total number of possible outcomes}} \)

This basic principle applies across all types of probability scenarios, including those involving dependent events.

Conditional probability takes the concept of probability a step further by considering how one event impacts the chance of another event occurring. Specifically, it is the probability of an event happening given that another event has already occurred.
To calculate conditional probability, we use the formula:

• \( P(B|A) = \frac{P(A \text{ and } B)}{P(A)} \)

Here, \( P(B|A) \) is the probability of event B occurring provided that event A has already happened. This is crucial for understanding dependent events, where outcomes influence subsequent possibilities.

Calculating the probability of dependent events requires understanding how one outcome affects another. In such situations, the probability of multiple events occurring is not merely the product of their separate probabilities because the events are linked.
To find the probability of two dependent events (e.g., event A and event B) occurring together, use:

• \( P(A \text{ and } B) = P(A) \times P(B|A) \)

Let's go over an example with marbles to see this in action.
Imagine you have a bag with 5 red balls and 3 green balls. If you draw one red ball and do not replace it, the probabilities shift because the sample space has changed.
Initially, the probability of drawing a red ball is \( \frac{5}{8} \). After removing one red ball, you have 4 red balls left out of 7 total balls. Therefore, the probability of drawing another red ball is now \( \frac{4}{7} \).
To find the combined probability of drawing two red balls consecutively, multiply their dependent probabilities: \( \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14} \).
This calculation illustrates how the first event of drawing affects the second, showing the interplay of probabilities in dependent situations.