Combined-Event Probability
- Combined-event probability is about finding the chance that two or more events happen together, or that at least one of them happens.
- In this topic you will use set notation, Venn diagrams, and tree diagrams, which we covered earlier, to represent situations and calculate probabilities clearly.
The Addition Rule Avoids Double Counting
- Suppose you are given two events $A$ and $B$.
- When you add $P(A)$ and $P(B)$, the overlap $A\cap B$ gets counted twice (once in $A$ and once in $B$).
- The addition rule fixes that: for any events $A$ and $B$,
$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ - If $A$ and $B$ are mutually exclusive, then $P(A\cap B)=0$ and the rule simplifies to: $$P(A\cup B)=P(A)+P(B)$$
Die roll (mutually exclusive events)
Roll a fair six-sided die.
- $A$: "roll a 1" so $P(A)=\frac16$
- $B$: "roll a 2" so $P(B)=\frac16$
You cannot roll 1 and 2 at the same time, so $P(A\cap B)=0$.
$$P(A\cup B)=\frac16+\frac16-0=\frac26=\frac13.$$
- Do not assume events are mutually exclusive just because they are different descriptions.
- For example, "roll an even number" and "roll a number greater than 3" do overlap (4 and 6).
Independent Events And The Multiplication Idea
Sometimes one event does not affect the probability of the other.
Independent events
Events $A$ and $B$ are independent if knowing that $A$ happened does not change the probability of $B$. Formally, $P(B\mid A)=P(B)$.
A common example is rolling a die twice: the first roll does not influence the second roll.
Multiplication Rule (For Independent Events)
- If $A$ and $B$ are independent, then $$P(A\cap B)=P(A)\,P(B)$$
- This connects to sample-space thinking: if outcomes combine in pairs, the "and" probability is the fraction of pairs that satisfy both conditions.
- Roll a 4-sided die (1–4) and flip a fair coin (H/T).
- The sample space has 8 equally likely outcomes: $$(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T)$$
- Probability of "roll a 1 and flip tails" is $\frac18$.
- Also, $P(\text{roll 1})=\frac14$ and $P(\text{tails})=\frac12$, so $$P(\text{roll 1 and tails})=\frac14\cdot\frac12=\frac18.$$
In MYP problems, independence is often supported by the context: separate trials (two rolls, roll + coin flip) are usually independent unless stated otherwise (for example, drawing without replacement).
Tree Diagrams Organize Multi-Step Experiments
- A tree diagram lists outcomes stage by stage.
- It helps you calculate combined probabilities by multiplying along a path and then, if needed, adding across paths.
How To Use A Tree Diagram
- Draw branches for the outcomes of the first event and label their probabilities.
- From each branch, draw the outcomes of the second event, again with probabilities.
- Multiply along branches to get the probability of each combined outcome.
- Add the relevant combined outcomes if the question asks for "or".
Odd on both dice
- For one fair die, $P(\text{odd})=\frac36=\frac12$.
- Two rolls are independent, so $$P(\text{odd on both})=\frac12\cdot\frac12=\frac14$$
- A tree diagram shows the same result by multiplying along the (Odd, Odd) path.
Conditional Probability Describes "Given That"
Sometimes the probability changes because you have extra information.
Conditional probability
The conditional probability of $B$ given $A$ is
$$P(B\mid A)=\frac{P(A\cap B)}{P(A)}, \quad P(A)>0.$$
This formula matches a simple idea: if you already know $A$ happened, you restrict attention to outcomes inside $A$.
- In a class, suppose $P(F)=0.4$, $P(M)=0.5$, and $P(F\cap M)=0.2$.
- Then $$P(M\mid F)=\frac{0.2}{0.4}=0.5$$
- Here, knowing a student studies French does not change the probability they study Mandarin, so these events would be independent in this data set.
- A common mistake is to write $P(A\mid B)=\frac{P(A)}{P(B)}$.
- The numerator must be the intersection $P(A\cap B)$.
Choosing A Representation And Checking Reasonableness
Different representations suit different problems:
- Lists/tables of outcomes are best for small sample spaces.
- Venn diagrams are best for overlap between categories (survey data).
- Tree diagrams are best for multi-stage experiments (several steps).
Two fast checks for combined-event answers:
- Range check: probabilities must be between 0 and 1.
- Overlap check: when using $P(A\cup B)$, ask yourself, "What outcomes are counted twice?"
- If there is an overlap, you must subtract $P(A\cap B)$.
- Explain in words the difference between $A\cup B$ and $A\cap B$.
- When can you simplify $P(A\cup B)$ to $P(A)+P(B)$?
- If $P(A)=0.7$, $P(B)=0.5$, and $P(A\cap B)=0.4$, compute $P(A\cup B)$.
- In your own example, describe two independent events and two events that are not independent.
