Probability:
counting what you want against what can happen

The basic formula

A fair six-sided die can land on six faces, each as likely as any other. One face shows a 5. So the chance of rolling a 5 is 1 out of 6, which is 0.1667, or 16.7% when written as a percentage.

That is the classical definition of probability. It always sits between 0, an impossible event, and 1, a certain one. It only works when the outcomes counted are equally likely, which is an assumption about the situation and not something the arithmetic can check for you.

Write it as a fraction first and convert at the end: 1/6 becomes 0.167 as a decimal, or 16.7% as a percentage. Keeping fractions through multi-step working avoids rounding drift.

Worked example: a total of 7 with two dice

Two fair dice have 6 × 6 = 36 equally likely ordered outcomes. A total of 7 can arise in six ways: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1.

Counting ordered pairs matters. If you counted 3+4 and 4+3 as one outcome, you would count 21 outcomes, and those are not equally likely.

Combining events: AND, OR and NOT

Notice the overlap term. Adding 13 hearts and 4 kings counts the king of hearts twice, so one is subtracted.

At least one: use the complement

Finding the chance of at least one six in four rolls by adding cases is messy. It is much easier to find the chance of no sixes and subtract from 1.

• Chance of no six on one roll: 5/6.
• Chance of no six in four independent rolls: (5/6)⁴ = 0.482.
• Chance of at least one six: 1 − 0.482 = 0.518, or 51.8%.

That is a little better than evens, a result that surprised gamblers in the seventeenth century and helped start the formal study of probability.

Odds versus probability

Probability and odds describe the same uncertainty in different ways, and mixing them up causes a surprising number of errors. Probability compares the favourable outcomes with all outcomes. Odds compare favourable with unfavourable.

To convert, odds of a to b in favour correspond to a probability of a ÷ (a + b). Bookmakers quote odds against, and the implied probability of 5 to 1 against is 1 ÷ 6, the same 16.7% as the die.

When the outcomes are not equally likely

The favourable-over-total rule breaks down when outcomes have different chances. A weather forecast of rain tomorrow is not 1 in 2 just because there are two outcomes. For such cases probability is estimated from data as a relative frequency, the number of times something happened divided by the number of trials.

If 36 of the last 120 deliveries to an address arrived late, the estimated probability of lateness is 36 ÷ 120 = 0.30. With more history the estimate becomes steadier; with only ten deliveries, one more late arrival moves it by about ten percentage points. This is the idea behind the law of large numbers: relative frequencies settle towards the true probability as trials accumulate.

• Classical probability: counted from symmetry, such as dice, coins and cards.
• Empirical probability: measured from observed frequencies.
• Subjective probability: a reasoned degree of belief when there is no repeatable experiment.

From probability to expected value

Probability becomes decision-making when you attach values to outcomes. Expected value is the sum of each outcome's value times its probability. A ticket costing ₹100 with a 1 in 1,000 chance of winning ₹50,000 has an expected return of 50,000 ÷ 1,000 = ₹50, half the price. It does not forecast any single draw, but it describes the average over a very large number of tickets.

The same logic underlies insurance pricing, quality-control sampling and medical screening. In screening, remember that a test's accuracy and the chance a positive result is correct are different probabilities. If a condition affects 1 person in 1,000 and a test is 99% accurate in both directions, then among 1,000 people about 1 has the condition and about 10 healthy people also test positive, so a positive result is right only around 9% of the time.

Whenever a probability looks counterintuitive, rewrite it as counts out of a fixed population, as above. The arithmetic becomes simple and the surprise usually disappears.

Without replacement

When events are not independent, the later probabilities change. Drawing two aces from a 52-card deck without putting the first back: the first draw is 4/52, then 3 aces remain in 51 cards, so the second is 3/51.