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Introduction to Probability
A bag has 3 red, 5 blue, and 2 green marbles. What is the probability of drawing a blue marble?
Learning Intentions
Know
- Sample space
- Event
- Probability scale
- Complementary events
Understand
- Why probabilities sum to 1
- The difference between theoretical and experimental probability
Can Do
- List a sample space
- Calculate simple probabilities
- Find complementary probabilities
Probability Basics
Probability measures how likely an event is to occur.
$P(E) = dfrac{ ext{number of favourable outcomes}}{ ext{total number of outcomes}}$
Probabilities range from 0 (impossible) to 1 (certain).
Example: Rolling a fair die, P(rolling a 4) = 1/6.
Sample Space
The sample space is the set of all possible outcomes of an experiment.
Example: Tossing a coin twice:
Sample space = {HH, HT, TH, TT}
Each outcome is equally likely if the coin is fair.
For a standard die: Sample space = {1, 2, 3, 4, 5, 6}
Complementary Events
The complement of event $E$ (written $E'$ or $E^c$) is the event that $E$ does not occur.
$P(E') = 1 - P(E)$
Example: P(rolling a 6) = 1/6, so P(not rolling a 6) = 5/6.
This is useful when $P(E)$ is easier to find than $P(E')$.
Check Understanding
A card is drawn from a standard 52-card deck. Find P(heart), P(king), and P(not a king).
Basic Probability
A bag has 4 red, 6 blue, and 10 green marbles. Find P(red), P(blue), P(green).
Total = 20
P(red) = 4/20 = 1/5
P(blue) = 6/20 = 3/10
P(green) = 10/20 = 1/2
A fair die is rolled. Find P(even number) and P(number greater than 4).
Even = {2, 4, 6}: P(even) = 3/6 = 1/2
Greater than 4 = {5, 6}: P(>4) = 2/6 = 1/3
Two coins are tossed. Find P(two heads) and P(at least one tail).
Sample space = {HH, HT, TH, TT}
P(two heads) = 1/4
P(at least one tail) = 1 - P(HH) = 1 - 1/4 = 3/4
Common Misconceptions
Add probabilities for all events to get more than 1. The sum of probabilities of all possible outcomes must equal 1. If your total exceeds 1, you have double-counted.
Assume all outcomes are equally likely. Only true for fair dice, coins, etc. Real-world situations often have unequal probabilities.
P(A) + P(B) = P(A or B) always. No — this only works for mutually exclusive events. If A and B can both occur, use the addition rule: P(A or B) = P(A) + P(B) − P(A and B).
Practice — Probability Basics
Gambling and Insurance
Insurance companies use probability to calculate premiums. The probability of car accidents for different age groups informs premium pricing. Australian lottery odds (e.g., Oz Lotto) are calculated using combinatorial probability — the chance of winning Division 1 is about 1 in 45 million.
📓 Copy Into Your Books
▼Probability
- $P(E) = ext{favourable}/ ext{total}$
- Range: 0 to 1
Sample space
- List all outcomes
- Check equally likely
Complement
- $P(E') = 1 - P(E)$
- Use when complement is easier