Topic 2.7 Notes – Independent Events and Unions of Events

What Independence and Union Mean

A quick symbol refresher helps a lot here:

• $A \cap B$ means both events happen.
• $A \cup B$ means $A$ or $B$ or both.
• $A^c$ means not $A$.

This Venn diagram is a good picture of union and intersection.

Venn diagram for $A \cup B$ and $A \cap B$

Independence is about whether probabilities change. If knowing $B$ happened does not change the chance of $A$, then the events are independent.

Equivalent ways to show independence:

$$P(A\mid B)=P(A)$$

$$P(B\mid A)=P(B)$$

$$P(A\cap B)=P(A)P(B)$$

A union is different. It just combines outcomes. It asks for the probability that at least one event occurs. So keep the ideas separate in your head:

• independence = relationship between events
• union = event made by combining outcomes

The Probability Rules You Need

For any two events, the addition rule is

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

That subtraction matters because the overlap gets counted twice if you just add.

Example. If $P(A)=0.38$, $P(B)=0.24$, and $P(A\cap B)=0.09$, then

$$P(A\cup B)=0.38+0.24-0.09=0.53$$

If events are independent, the intersection can be found by multiplying:

$$P(A\cap B)=P(A)P(B)$$

Only do that if independence is given or proven.

A few special cases get tested a lot:

• Mutually exclusive
  • They cannot happen together, so $P(A\cap B)=0$
  • Then $P(A\cup B)=P(A)+P(B)$
• Neither event
  • $P(\text{neither})=1-P(A\cup B)$
• Exactly one event
  • $P(A\cup B)-P(A\cap B)$
  • equivalent form is $P(A)+P(B)-2P(A\cap B)$

If events are independent, you can plug the product into the addition rule:

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

How to Choose the Right Rule

The wording tells you the event. The relationship tells you the formula.

1. Translate the words.
   • both $\rightarrow A\cap B$
   • or / either / at least one $\rightarrow A\cup B$
   • neither $\rightarrow (A\cup B)^c$
   • exactly one $\rightarrow$ union without overlap
2. Decide what relationship you know.
   • independent $\rightarrow$ use multiplication for the intersection
   • mutually exclusive $\rightarrow$ intersection is $0$
   • neither given $\rightarrow$ do not assume

For “at least one” with independent events, the complement is often cleaner:

$$P(A\cup B)=1-P(A^c\cap B^c)=1-[1-P(A)][1-P(B)]$$

Example. If $P(A)=0.3$ and $P(B)=0.4$, then

$$P(\text{at least one})=1-(0.7)(0.6)=0.58$$

On an FRQ, show the event, the rule, the substitution, and the final probability in context.

How to Check Independence in Problems

You may need to prove independence instead of being told.

Given probabilities

Check either of these:

• $P(A\cap B)=P(A)P(B)$
• $P(B\mid A)=P(B)$

Equally likely outcomes

On one die roll, let $A=\{\text{even}\}$ and $B=\{>4\}$.

• $P(A)=\frac{3}{6}=\frac12$
• $P(B)=\frac{2}{6}=\frac13$
• $P(A\cap B)=\frac{1}{6}$

Since $\frac12\cdot\frac13=\frac16$, the events are independent.

Two-way table

Use:

• marginal proportion for $P(A)$ and $P(B)$
• joint proportion for $P(A\cap B)$
• conditional proportion for $P(B\mid A)$

Use exact counts when possible. Rounding can fake a mismatch.

Separate trials are often independent. Without replacement usually makes events dependent.

What Students Mix Up

The biggest trap is independent vs. mutually exclusive.

• Mutually exclusive means they cannot both happen.
• Independent means one does not affect the other.

If both events have positive probability, they cannot be both mutually exclusive and independent.

Other common misses:

• using $P(A)+P(B)$ when events overlap
• multiplying $P(A)P(B)$ without knowing independence
• forgetting that “or” includes the overlap
• treating “at least one” and “exactly one” as the same
• assuming same experiment means dependent, or sequence means independent