4.2.4. Addition and Multiplication Theorems

• Addition Theorem of Probability: This theorem helps us calculate the probability of the occurrence of either of two events, denoted as 𝐴 and 𝐵, as:

$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$

Here: • 𝑃(𝐴∪𝐵) is the probability of event 𝐴 or event 𝐵 occurring. • 𝑃(𝐴∩𝐵) is the probability of both events occurring.

The Addition Theorem of Probability is a fundamental concept in probability theory. It provides a way to find the probability of at least one of two events happening. To apply this theorem, you need to know the probabilities of event A and event B. The formula states that the probability of either event A or event B occurring is equal to the sum of their individual probabilities minus the probability that both events occur at the same time. This subtraction is necessary to avoid double counting the case where both events happen.

Imagine you have a bag with 5 red balls and 3 blue balls. If you want to find the probability of randomly pulling out either a red or a blue ball, you first find the probability of pulling a red ball (5/8) and the probability of pulling a blue ball (3/8). If you add these probabilities, you would get 1 (or 100%), which is incorrect since the two colors are not mutually exclusive. In this example, you don't need to subtract anything as those colors cannot simultaneously happen when picking one ball.

• Multiplication Theorem of Probability: This theorem gives the probability of the simultaneous occurrence of two events. For independent events 𝐴 and 𝐵, the probability of both events occurring is:

$P(A ∩ B) = P(A) × P(B)$

If the events are dependent, the formula adjusts to account for conditional probability.

The Multiplication Theorem of Probability focuses on determining the probability that two events happen at the same time. When the events are independent, meaning the occurrence of one does not affect the occurrence of the other, you can simply multiply the probabilities of the two events. However, if the events are dependent, meaning one event affects the likelihood of the other occurring, you need to adjust the formula by using conditional probabilities, which are probabilities that take into account that one event has already occurred.

Consider this: You roll a die and flip a coin. The chance of rolling a 4 on the die is 1/6, and the chance of getting heads on the coin is 1/2. Since the outcome of the die does not affect the outcome of the coin, these events are independent. To find the probability of rolling a 4 and getting heads, you multiply the individual probabilities: (1/6) × (1/2) = 1/12. In contrast, if you were drawing cards from a deck without replacement, the probability of drawing a second card would depend on what the first card was, showcasing dependency.

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