The reason we subtract $P(X \cap Y)$ when calculating $P(X \cup Y)$ is due to the principle of inclusion-exclusion in probability theory. Here's a detailed explanation:

Inclusion-Exclusion Principle

When we calculate $P(X \cup Y)$, we are looking for the probability that either event $X$ or event $Y$ or both occur. The formula for the union of two events is:

$P(X \cup Y) = P(X) + P(Y) - P(X \cap Y)$

Why Subtract $P(X \cap Y)$?

1. Double Counting:

   • When we add $P(X)$ and $P(Y)$ together, we count the probability of the events occurring independently.
   • However, if there is an overlap between events $X$ and $Y$ (i.e., $X \cap Y$), we count this overlap twice.

2. Correcting the Overlap:

   • To avoid this double counting, we need to subtract the probability of the intersection $P(X \cap Y)$.
   • This subtraction ensures that the overlapping part is only counted once.

Example

Let's consider a concrete example for clarity:

• Suppose we have two events: $X$ (rain) and $Y$ (umbrella).
• Let's say $P(X) = 0.4$ (there's a 40% chance of rain), $P(Y) = 0.5$ (there's a 50% chance of carrying an umbrella), and $P(X \cap Y) = 0.2$ (there's a 20% chance of rain and carrying an umbrella at the same time).

Using the formula for the union of two events:

$P(X \cup Y) = P(X) + P(Y) - P(X \cap Y)$ $P(X \cup Y) = 0.4 + 0.5 - 0.2$ $P(X \cup Y) = 0.7$

Here’s a breakdown of what’s happening:

• Adding $P(X)$ and $P(Y)$ gives us 0.9.
• This 0.9 includes the intersection (rain and umbrella) twice.
• Subtracting $P(X \cap Y)$ (0.2) corrects for this double counting, resulting in the correct probability of either rain or carrying an umbrella or both, which is 0.7.

Conclusion

Subtracting $P(X \cap Y)$ is necessary to account for the overlap between events $X$ and $Y$ to ensure we don't double count the intersection.

Would you like more details or have any questions?

Here are some related questions for further practice:

1. How would you extend the inclusion-exclusion principle to three events $X$, $Y$, and $Z$?
2. What happens if the events $X$ and $Y$ are mutually exclusive? How does the formula simplify?
3. How can the principle of inclusion-exclusion be applied to solve problems involving more than two events?
4. What is the probability of the complement of the union of two events, $P((X \cup Y)^c)$?
5. How do you interpret the results of $P(X \cup Y)$ in a real-world context, like weather forecasting?
6. Can you derive the formula for the union of two events using Venn diagrams?
7. What is the relationship between the union and intersection of two independent events?
8. How do conditional probabilities change the calculation of the union of two events?

Tip: Visual aids like Venn diagrams can be very helpful in understanding and solving problems involving unions and intersections of events.