Probability Starts With A Sample Space
- A probability experiment is any repeatable situation with uncertain outcomes, for example tossing a coin, rolling a die, or picking a raffle ticket.
- To talk about probability clearly, we first describe all possible outcomes.
Sample space
The set of all possible outcomes, often written $U$ or $S$.
A sample space can be written in different ways:
- List: rolling a standard die: $S=\{1,2,3,4,5,6\}$
- Table: sums from rolling two dice (each cell is an outcome)
- Tree diagram: outcomes that happen in stages (first coin toss, then second toss, etc.)
The outcomes you include must match the experiment exactly.
"Rolling two dice" needs outcomes like $(2,5)$ and $(5,2)$ separately (because they are different ordered outcomes), even though they give the same sum.
- A very common mistake is to forget outcomes or to treat outcomes as equally likely when they are not.
- Always define the experiment first, then build the sample space.
Events Are Subsets Of The Sample Space
Once you have $S$, you can define what you care about.
Event
A set of outcomes from a random experiment (for example, “rolling an odd number”).
Complement
The set of elements not in a set. The complement of $A$ (relative to $U$) is $A'$.
Given one die: $$S=\{1,2,3,4,5,6\}$$
- Let $A$ be the event "getting a 3".
- Then $A=\{3\}$.
- The event "not getting a 3" is called the complement.
- This implies $A'=\{1,2,4,5,6\}$.
- Because either $A$ happens or it does not, $A \cup A' = S$.
The Single-Event Probability Formula
- For single-event probability in a situation where outcomes are equally likely, we use: $$\mathrm{P}(A)=\frac{\text{number of outcomes in }A}{\text{number of outcomes in }S}=\frac{n(A)}{n(S)}$$
- This is the key idea: probability is a proportion of the sample space.
In real life, probability can also be interpreted as the proportion of times an event would occur over a very large number of trials.
Axioms You Must Always Obey
Probability has three basic rules (axioms):
- Non-negativity: For any event $A$, $\mathrm{P}(A)\ge 0$.
- Total probability: $\mathrm{P}(S)=1$.
- Addition for mutually exclusive events: If events $A_1, A_2, \dots$ cannot happen together, then $$\mathrm{P}(A_1\cup A_2\cup \cdots)=\mathrm{P}(A_1)+\mathrm{P}(A_2)+\cdots$$
Mutually exclusive
Two events are mutually exclusive if they cannot happen at the same time. In set notation, $A\cap B=\varnothing$, so $\mathrm{P}(A\cap B)=0$.
Using Complements To Calculate A Probability
- The complement rule is extremely useful: $$\mathrm{P}(A)+\mathrm{P}(A')=1$$
- So, $$\mathrm{P}(A')=1-\mathrm{P}(A)$$
Given one die:
$$\mathrm{P}(\text{get a 3})=\frac{1}{6}$$
$$\mathrm{P}(\text{not get a 3})=1-\frac{1}{6}=\frac{5}{6}$$
- If an event is described as "not …", try using the complement.
- It can turn a hard count into an easy one.
Building Sample Spaces In Common Formats
Lists For Simple Experiments
- If you toss a coin once: $S=\{H,T\}$.
- If you roll a 4-sided die: $S=\{1,2,3,4\}$.
Tables For Two-Stage Number Experiments
- If you roll two 4-sided dice (faces $1$ to $4$), the sample space has $4\times 4=16$ equally likely ordered outcomes.
- A table helps you see sums or products.
- For example, the event "sum is 5" happens for $(1,4),(2,3),(3,2),(4,1)$, so: $$\mathrm{P}(\text{sum }=5)=\frac{4}{16}=\frac{1}{4}$$
Lists For Choice Experiments
- If a street vendor sells two sandwiches (Gatsby, Gyro) and three fruits (apple, orange, banana), then there are $2\times 3=6$ outcomes: $$S=\{(\text{Gatsby, apple}), (\text{Gatsby, orange}), (\text{Gatsby, banana}), $$ $$(\text{Gyro, apple}), (\text{Gyro, orange}), (\text{Gyro, banana})\}$$
- An event could be "choosing a banana", which has 2 outcomes out of 6, so probability $=\frac{2}{6}=\frac{1}{3}$.
- If 500 tickets are sold and exactly one ticket wins, and you buy $k$ tickets, then the sample space has 500 equally likely tickets.
- Event $W$ = "you win".
- There are $k$ winning possibilities for you (any of your tickets could be the winner), so: $$\mathrm{P}(W)=\frac{k}{500}$$
- If you buy 10 tickets, $\mathrm{P}(W)=\frac{10}{500}=\frac{1}{50}=0.02$.
- If you buy 0 tickets, $\mathrm{P}(W)=0$ (impossible).
- If you buy all 500 tickets, $\mathrm{P}(W)=1$ (certain).
A raffle has 200 tickets. You buy 15. What is the probability you win? What is the probability you do not win?
Tree Diagrams For Single Events In Multi-Step Games
- A tree diagram is a structured way to list outcomes when an experiment happens in stages.
- Each branch shows an outcome and its probability.
The Two-Player Coin Game
Rules:
- Peter flips first. If he gets Head, Peter wins. If Tail, Eliott flips.
- Eliott flips. If Head, Eliott wins. If Tail, it returns to Peter.
This is a multi-step process, but each question like "Peter wins on his first turn" is still a single event (one event you are calculating).
(i) Probability Peter Wins On His First Turn
Peter wins immediately if his first flip is Head: $$\mathrm{P}(\text{Peter wins on first turn})=\frac{1}{2}$$
(ii) Probability Eliott Wins On His First Turn
For Eliott to win on his first turn, Peter must first flip Tail, then Eliott flips Head:
$$\mathrm{P}(T \text{ then } H)=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$$
(iii) Probability Peter Wins On His Third Turn
Peter's third turn happens only if the first four flips are all tails (Peter T, Eliott T, Peter T, Eliott T), and then Peter flips Head on his third turn.
That path is: $T,T,T,T,H$.
$$\mathrm{P}(TTTTH)=\left(\frac{1}{2}\right)^5=\frac{1}{32}$$
- On a tree diagram, multiply along a path (AND).
- If you have several different winning paths, add the path probabilities (OR), but only if those paths are mutually exclusive.
Fairness And What "Equal Chance" Really Means
- A fair game usually means each player has the same probability of winning.
- In the coin game from the example above, Peter always goes first, so Peter has an advantage.
- You can already see it because:
- Peter can win immediately with probability $\frac{1}{2}$.
- Eliott only gets a chance to win if Peter fails first.
- If you continued the tree, Peter's overall chance of winning is greater than Eliott's.
- Going first in this game is like getting the first attempt at a multiple-choice question where the first correct answer ends the round.
- Even if later players have the same chance per attempt, they get fewer opportunities before the round ends.
Designing a fair game often means balancing "who goes first" with the rules, for example by changing the win condition or giving the second player a compensating advantage.
Simulation As A Reality Check
- Sometimes you are told a probability (for example, "a winning envelope occurs 1 in 6 times").
- One way to test how believable results are is to simulate.
- If the chance is $\frac{1}{6}$, it matches "rolling a 6 on a fair die".
- If 8 customers open envelopes, you can simulate by rolling a die 8 times and counting sixes.
- Small samples can look surprising.
- Getting 3 wins out of 8 does not automatically prove the claim is wrong, because random variation is expected in short runs.
- Not writing the sample space (or writing an incomplete one).
- Assuming all outcomes are equally likely when they are not.
- Adding probabilities when you should multiply (AND vs OR confusion).
- Forgetting that $\mathrm{P}(A)+\mathrm{P}(A')=1$.
- Write the sample space for tossing a coin and rolling a die at the same time.
- Find $\mathrm{P}(\text{tail and an even number})$.
- Are the events "tail" and "even number" mutually exclusive? Explain.
