Probability

Q: You roll one die. What is the probability of getting an even number?

1/2 — The even faces are 2, 4 and 6 — that is 3 outcomes out of 6. So P = 3/6 = 1/2.

Q: One coin lands heads with probability 1/2. If you flip two coins, what is the probability that both land heads?

1/4 — The two flips are independent, so multiply: 1/2 × 1/2 = 1/4.

Q: You roll two dice. What is the probability that the total is 8?

5/36 — Five of the 36 outcomes total 8: (2,6), (3,5), (4,4), (5,3), (6,2). So the probability is 5/36.

Q: A bag holds 3 red balls and 2 blue balls. You take one out without looking. What is the probability it is blue?

2/5 — There are 5 balls in total and 2 of them are blue, so P = 2/5.

Q: Flip two coins. Using the complement, what is the probability of getting at least one head?

3/4 — The opposite of "at least one head" is "both tails", which has probability 1/4. So the answer is 1 − 1/4 = 3/4.

Q: An item has a 3% drop rate. What is the probability that a single pull does NOT give it to you?

97% — Missing is the complement of hitting: 1 − 0.03 = 0.97, or 97%.

Q: You have one red, one blue and one yellow card. You draw two of them one after another. What is the probability that the red card comes first?

1/3 — Only the first draw matters here: 1 red card out of 3, so the probability is 1/3.

Q: The chance of rain is 30% on each of three days in a row. What is the probability that it rains on all three days?

2.7% — Treat the days as independent and multiply: 0.3 × 0.3 × 0.3 = 0.027, which is 2.7%.

Where this is used

"Does this medicine actually work?" "Is this insurance price fair?" "What are my odds of pulling the rare item?" Every one of those questions is answered with probability. Insurance companies price risk by putting a number on how likely something is, drug companies run trials to show an effect is bigger than luck, and game studios tune loot-box odds to hit a revenue target. Probability is the tool that turns an uncertain world into numbers you can argue about.

Start gently

Probability turns "how likely" into a number between 0 and 1. When you have to count every possible case, drawing a tree stops you missing any.

Key points

What a Probability Is

If there are n outcomes that are all equally likely, and a of them count as event A, then P(A) = a/n. A probability is always between 0 and 1: 0 means it never happens, 1 means it always does.

The Complement: "Everything Else"

The event "A does not happen" is called the complement, written Aᶜ, and P(Aᶜ) = 1 − P(A). Whenever a question says "at least one...", it is usually far quicker to work out the chance of none and subtract from 1.

Counting with Two Dice

Draw a 6 × 6 grid for the two dice and you get 36 equally likely outcomes. Six of them add up to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So P = 6/36 = 1/6.

Independent Events Multiply

When one event has no effect on the other, you multiply to find the chance of both happening. Flip two coins and ask for two heads: P = 1/2 × 1/2 = 1/4.

What a 1% Drop Rate Really Costs

If an item has a 1% drop rate, the average number of pulls you need is 1 ÷ 0.01 = 100. At ¥300 a pull, that is 100 × ¥300 = ¥30,000 of expected spending — and "average" means plenty of players pay far more.

See it drawn

Tossing a coin twice (4 outcomes in all)

The chance of at least one head is 3/4. Counting the cases gets you there with no formula at all.

Check yourself 10questions

Tap an option to check

Q1 You roll one die. What is the probability of getting an even number?

Q2 One coin lands heads with probability 1/2. If you flip two coins, what is the probability that both land heads?

Q3 You roll two dice. What is the probability that the total is 8?

Q4 A bag holds 3 red balls and 2 blue balls. You take one out without looking. What is the probability it is blue?

Q5 Flip two coins. Using the complement, what is the probability of getting at least one head?

Q6 An item has a 3% drop rate. What is the probability that a single pull does NOT give it to you?

Q7 You have one red, one blue and one yellow card. You draw two of them one after another. What is the probability that the red card comes first?

Q8 The chance of rain is 30% on each of three days in a row. What is the probability that it rains on all three days?

Q9 A box holds 10 tickets, 3 of which are winners. You draw two tickets in a row without putting the first one back. What is the probability that both are winners?

Q10 Which statement about probability is true?

Jobs that use this

Actuary $180k

Game Designer (Monetization) $90k

Biostatistician $100k

Actuary$180k

Uses probability and statistics on death, accident and illness rates to price insurance policies and design pension plans. The qualifying exams are among the hardest in any profession.

Maths used: ProbabilityStatisticsCalculusStochastic processes

Game Designer (Monetization)$90k

Sets drop rates, loot box odds and pity timers, shaping how and when players decide to spend — all of it built on probability.

Maths used: ProbabilityStatisticsExpected valueBehavioral psychology

Biostatistician$100k

Analyzes clinical trial data with probability and statistics to prove scientifically that a drug is both effective and safe.

Maths used: ProbabilityStatisticsHypothesis testingBayesian inference