Additive Rules

Additive Rules in Probability

The additive rules of probability describe how to calculate the probability of the union of two or more events. These rules help determine the probability that at least one of several events occurs. There are two key additive rules: the General Addition Rule and the Special Addition Rule. Let's explore both in detail.

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1. Special Addition Rule (for Mutually Exclusive Events)

When two events are mutually exclusive (also known as disjoint), it means that they cannot both happen at the same time. For example, when rolling a die, the event of rolling a 2 and the event of rolling a 5 are mutually exclusive because you cannot roll both a 2 and a 5 in a single throw.

Rule:

If two events $A$ and $B$ are mutually exclusive, then the probability of either event occurring is the sum of their individual probabilities:

$$P(A \cup B) = P(A) + P(B)$$

Where:

• $P(A \cup B)$ is the probability that event $A$ or event $B$ occurs.
• $P(A)$ is the probability of event $A$.
• $P(B)$ is the probability of event $B$.

Example: Rolling a Die

• Event $A$: Rolling a 2.
• Event $B$: Rolling a 5.
• These two events are mutually exclusive, because you cannot roll both a 2 and a 5 on the same die at the same time.

$$P(A) = \frac{1}{6}, \quad P(B) = \frac{1}{6}$$

So, using the Special Addition Rule:

$$P(A \cup B) = P(A) + P(B) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$$

Thus, the probability of rolling either a 2 or a 5 is $\frac{1}{3}$.

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2. General Addition Rule (for Non-Mutually Exclusive Events)

If the events are not mutually exclusive, meaning that both events can happen at the same time (they have an overlap), we need to adjust for the fact that we may have counted the overlapping outcomes twice.

Rule:

For two events $A$ and $B$, the probability of either event occurring is:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Where:

• $P(A \cup B)$ is the probability that either event $A$ or event $B$ occurs.
• $P(A)$ is the probability of event $A$.
• $P(B)$ is the probability of event $B$.
• $P(A \cap B)$ is the probability that both events $A$ and $B$ occur at the same time (i.e., the intersection of $A$ and $B$).

Why subtract $P(A \cap B)$?

The reason we subtract $P(A \cap B)$ is that it is included in both $P(A)$ and $P(B)$. Without subtracting it, we would be double-counting the probability of the overlap.

Example: Drawing Cards from a Deck

• Event $A$: Drawing a red card (26 red cards in a standard deck of 52 cards).
• Event $B$: Drawing a face card (12 face cards in a standard deck of 52 cards).

These two events are not mutually exclusive because there are red face cards (i.e., the 6 red face cards: Jack, Queen, and King of Hearts and Diamonds).

$$P(A) = \frac{26}{52} = \frac{1}{2}, \quad P(B) = \frac{12}{52} = \frac{3}{13}$$

Now, the probability of drawing a red face card (the intersection of $A$ and $B$):

$$P(A \cap B) = \frac{6}{52} = \frac{3}{26}$$

Now, applying the General Addition Rule:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ $$P(A \cup B) = \frac{1}{2} + \frac{3}{13} - \frac{3}{26}$$

To simplify:

$$P(A \cup B) = \frac{13}{26} + \frac{6}{26} - \frac{3}{26} = \frac{16}{26} = \frac{8}{13}$$

Thus, the probability of drawing either a red card or a face card is $\frac{8}{13}$.

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3. Additive Rule for Three or More Events

The General Addition Rule can also be extended to more than two events. For three events $A$, $B$, and $C$, the rule is:

$$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$$

This rule can be extended for more events, but the key concept is the same: adding probabilities of individual events and subtracting the probabilities of their intersections to avoid double-counting.

Example: Rolling Three Dice

• Event $A$: Rolling a 1 on the first die.
• Event $B$: Rolling a 1 on the second die.
• Event $C$: Rolling a 1 on the third die.

These events are not mutually exclusive, as multiple dice can show 1s simultaneously. To find the probability of rolling a 1 on at least one die, you would apply the General Addition Rule for three events.

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Summary of Additive Rules

• Special Addition Rule (for Mutually Exclusive Events):

  $$P(A \cup B) = P(A) + P(B)$$

  Use this when the events cannot happen at the same time (e.g., rolling a 2 or a 5 on a die).

• General Addition Rule (for Non-Mutually Exclusive Events):

  $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

  Use this when the events can happen at the same time (e.g., drawing a red card or a face card from a deck).

• Addition Rule for More Than Two Events: The general rule extends to more events and requires subtracting intersections and adding back the intersection of all events.

The additive rules of probability are fundamental for calculating the likelihood of at least one of several events occurring, especially when events are not mutually exclusive.