Probability lessons

Likelihood as a fraction

When every outcome is equally likely, probability is a fraction. Count the outcomes you want and divide by the total number of outcomes.

P(A) = \frac{\text{favourable outcomes}}{\text{total outcomes}}

For a fair die, the chance of rolling a 4 is \frac{1}{6} because there is one 4 out of six equally likely faces.

The probability scale

Probabilities can be written as fractions, decimals, or percentages. All three forms describe the same likelihood.

Drag event cards onto a 0-to-1 probability number line.

Practice: Calculate simple probabilities

Type the probability of drawing a colour from a bag of marbles as a fraction.

Sample spaces and outcomes

The sample space is the complete set of all possible outcomes. We write it as S or list it in curly brackets.

For a coin flip, S = \{H, T\}. For a die roll, S = \{1, 2, 3, 4, 5, 6\}.

An event is any subset of the sample space. "Rolling an even number" is the event \{2, 4, 6\}.

Listing outcomes systematically

When two things happen together, list every combination. For flipping a coin and rolling a die, there are 2 \times 6 = 12 outcomes.

Fill a coin-by-die outcome grid, then highlight an event and its probability.

Practice: Find the sample space

Type the number of outcomes and select those matching an event.

Theoretical vs experimental probability

Theoretical probability uses logic and counting. You calculate it before doing any experiment.

Experimental probability (or relative frequency) comes from actually running trials. It equals the number of times the event happened divided by the total trials.

P_{exp}(A) = \frac{\text{times A occurred}}{\text{total trials}}

Coin flip simulator

As you run more trials, experimental probability gets closer to the theoretical value. This is called the law of large numbers.

Flip a coin and watch experimental probability converge on 0.5.

Practice: Estimate from data

Estimate experimental probability from a frequency table, then an expected count.

Complementary events

The complement of an event A is everything in the sample space that is not A. We write it as A' or \bar{A}.

Since an event either happens or it does not, the two probabilities always add to 1.

Complement rule: P(A') = 1 - P(A)

Using complements as shortcuts

Sometimes it is easier to find the probability of what you do not want, then subtract from 1.

1. P(\text{at least one six in two rolls}) (find this)
2. = 1 - P(\text{no sixes in two rolls}) (use the complement)
3. = 1 - \frac{5}{6} \times \frac{5}{6} (each roll has 5/6 chance of not being six)
4. = 1 - \frac{25}{36} = \frac{11}{36} (result)

Slide P(A) and watch the complement fill the rest of the sample space.

Practice: Complement problems

Use the complement rule to answer probability problems.

Combined events: the OR rule

When you want the probability of event A or event B happening, you add their probabilities. But if they can both happen at the same time, you must subtract the overlap.

Addition rule: P(A \cup B) = P(A) + P(B) - P(A \cap B)

If A and B cannot happen together (mutually exclusive), the overlap is zero and the formula simplifies to P(A) + P(B).

Combined events: the AND rule

When you want both events to happen, you multiply. For independent events (one does not affect the other):

Multiplication rule (independent): P(A \cap B) = P(A) \times P(B)

Flipping heads then rolling a 6 gives \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}. The coin does not affect the die, so they are independent.

Venn diagrams for OR and AND

A Venn diagram shows two overlapping circles inside a rectangle. The overlap is A \cap B and the total shaded area is A \cup B.

Set two-circle Venn region probabilities and read off P(A or B) and P(A and B).

Practice: OR and AND problems

Apply the addition rule to find P(A or B) as a fraction.

Tree diagrams

A tree diagram shows sequences of events as branches. Each branch is labelled with its probability, and you multiply along a path to get the probability of that sequence.

To find the probability of any collection of paths, add the results of each path together.

Reading a tree diagram

Here is a tree for flipping a coin twice. Each path represents one complete outcome.

1. \textcolor{#f871b9}{P(HH)} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} (heads then heads)
2. \textcolor{#8b5cf6}{P(HT)} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} (heads then tails)
3. P(\text{at least one head}) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4} (add HH, HT, TH paths)

Click paths on a two-flip probability tree to see their multiplication.

Practice: Tree diagram calculations

Multiply along a tree diagram path to find a probability.

Independent and dependent events

Two events are independent if the outcome of one does not change the probability of the other. Flipping a coin twice gives independent events.

Events are dependent when the first outcome changes what is available for the second. Drawing cards without replacing them is a classic example.

Without replacement

When you do not put an item back, the total count drops and the probabilities shift on the second draw.

1. \textcolor{#f871b9}{P(\text{1st red})} = \frac{4}{10} (4 red out of 10 total)
2. \textcolor{#8b5cf6}{P(\text{2nd red})} = \frac{3}{9} (one red gone, one fewer total)
3. P(\text{both red}) = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15} (multiply along the path)

Draw marbles with and without replacement to see how the fractions change.

Practice: Independent or dependent?

Classify events as independent or dependent, then calculate the probability.

Conditional probability

Conditional probability is the probability of event A happening given that B has already happened. We write it as P(A | B).

It narrows the sample space down to only the outcomes where B is true, then counts how many of those also satisfy A.

Conditional probability: P(A | B) = \frac{P(A \cap B)}{P(B)}

Conditional probability in action

Imagine 100 students: 40 study maths, 30 study science, and 10 study both. Given a student studies maths, what is the probability they also study science?

1. P(\text{science} | \text{maths}) = \frac{P(\text{both})}{P(\text{maths})}
2. = \frac{10/100}{40/100} = \frac{10}{40} = \frac{1}{4} (restrict to the 40 maths students)

Edit a two-way table and compute conditional probabilities against a live Venn diagram.

Practice: Conditional probability

Read a two-way table to answer conditional probability questions.

Expected value

The expected value is the long-run average outcome of a random process. It tells you what to expect on average, not on any single trial.

Multiply each outcome by its probability and add them all up.

Expected value: E(X) = \sum x_i \cdot P(x_i)

Calculating expected value

A game costs £2 to play. You roll a die: if you roll a 6 you win £10, otherwise you win nothing. Is the game worth playing?

1. E(\text{winnings}) = 10 \times \frac{1}{6} + 0 \times \frac{5}{6} (multiply outcomes by probabilities)
2. = \frac{10}{6} \approx 1.67 (expected winnings per game)
3. \text{profit} = 1.67 - 2 = -0.33 (subtract the cost to play)

On average you lose about 33p per game. The game is not worth playing in the long run.

Roll a dice game and watch average winnings converge on the expected value.

Practice: Find the expected value

Calculate the expected value of a distribution and judge if a game is fair.

Probability distributions

A probability distribution lists every possible outcome alongside its probability. The probabilities must add up to exactly 1.

A uniform distribution means every outcome is equally likely. Rolling a fair die is uniform because each face has probability \frac{1}{6}.

The binomial distribution

The binomial distribution counts the number of successes in a fixed number of independent trials, each with the same probability of success.

If you flip a coin n times, the probability of getting exactly k heads is:

Binomial formula: P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Here p is the probability of success on one trial and \binom{n}{k} counts the ways to choose which k trials are successes.

Binomial distribution explorer

Changing n and p reshapes the distribution. Watch how the histogram shifts as you adjust the parameters.

Adjust n and p to reshape a binomial distribution histogram.

Practice: Binomial calculations

Use the binomial formula to find a probability to 3 decimal places.

The normal distribution

The normal distribution is a continuous, bell-shaped curve. It appears everywhere in nature because of how averages of random values behave.

It is defined by two numbers: the mean \mu (the centre) and the standard deviation \sigma (the spread).

Normal distribution notation: X \sim N(\mu, \sigma^2)

The 68-95-99.7 rule

For any normal distribution, about 68% of values fall within 1\sigma of the mean, 95% within 2\sigma, and 99.7% within 3\sigma.

1. P(\mu - \sigma < X < \mu + \sigma) \approx 0.68 (68% within 1 standard deviation)
2. P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.95 (95% within 2 standard deviations)
3. P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.997 (99.7% within 3 standard deviations)

Reshape a normal curve and explore the 68-95-99.7 rule bands.

Practice: Normal distribution questions

Pick the empirical-rule percentage for a shaded region of a normal curve.