How to find probability

The probability formula

P(A) is read "the probability of A", where A is the event you care about. To use the formula:

1. List or count all the possible outcomes. Make sure they are equally likely.
2. Count the outcomes where your event happens.
3. Divide the second number by the first, and simplify the fraction.

Fractions, decimals and percentages all work: 1/4, 0.25 and 25% are the same probability. Never write it as a ratio like 1:4, and never give a probability above 1 or below 0. If you get one, something has gone wrong.

Example 1: a bag of marbles

A bag holds 6 red, 2 blue and 4 green marbles. You take one without looking. What is the probability it is blue?

There are 6 + 2 + 4 = 12 marbles, all equally likely to be picked. Two of them are blue.

Example 2: playing cards

A standard pack has 52 cards: 4 suits (hearts, diamonds, clubs, spades) of 13 cards each. The probability of drawing a heart is 13/52 = 1/4, and the probability of drawing a king is 4/52 = 1/13.

How to do probability with two dice: list the sample space

When two things happen at once, list every possible outcome first. This list is called the sample space. For two dice, a grid works best: each cell is one outcome, and there are 6 × 6 = 36 of them, all equally likely. Each cell shows the total of the two dice.

Now count. A total of 7 appears in 6 cells, so P(total of 7) = 6/36 = 1/6, and 7 is the most likely total. A total of 11 or more appears in only 3 cells, so P(11 or more) = 3/36 = 1/12. A common mistake is to say there are 11 possible totals, from 2 to 12, so each has a probability of 1/11. The totals aren’t equally likely, but the 36 cells are.

The probability of something not happening

An event either happens or it doesn't, so the two probabilities add up to 1. That gives you a shortcut:

In the marble bag, P(green) = 4/12, so P(not green) = 1 − 4/12 = 8/12 = 2/3. You could also count the 8 marbles that aren't green, but when "not" covers lots of outcomes, subtracting from 1 is much quicker. "Not A" is called the complement of A and is often written A′ (A prime).

Finding the probability of A or B

If two events can't happen at the same time, they are mutually exclusive, and you add their probabilities. A marble can't be red and blue at once, so P(red or blue) = 6/12 + 2/12 = 8/12 = 2/3.

If they can happen together, adding counts the overlap twice, so take it away once:

For a heart or a king: 13 hearts plus 4 kings counts the king of hearts twice. So P(heart or king) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13. A Venn diagram makes this easy to see: the overlap of the two circles is the outcomes in both events. In set notation, "or" is written ∪ (union) and "and" is written ∩ (intersection).

Finding the probability of A and B

If one event doesn't change the chances of the other, they are independent, and you multiply:

A spinner has 5 equal sections, 2 of them yellow. You spin it and flip a coin. The spinner can't affect the coin, so P(yellow and heads) = 2/5 × 1/2 = 2/10 = 1/5.

A good way to remember which rule is which: "or" makes a probability bigger, so you add; "and" asks for more to go right, so the probability gets smaller, and multiplying two fractions less than 1 makes a smaller number.

Dependent events: without replacement

If the first event changes the second, the events are dependent. The classic case is picking twice without putting the first item back. A drawer has 4 white socks and 6 black socks. You pull out two in the dark. What is the probability both are white?

First sock: 4 of the 10 are white. If it is white, only 3 white socks are left out of 9. So:

That 3/9 is a conditional probability: the probability the second sock is white given the first was white, written P(second white | first white). Try one: this question asks whether two events are independent or dependent.

Using a tree diagram

A tree diagram sets out two or more events in a row. Each set of branches shows the outcomes of one event, with probabilities that add up to 1. To find the probability of a path, multiply along the branches. To combine several paths, add their results. Here is the sock drawer from above, with W for white and B for black.

The four paths add up to 90/90 = 1, a useful check. The probability of getting one of each colour uses two paths, white then black or black then white: 24/90 + 24/90 = 48/90 = 8/15. Our Probability course lets you build and explore trees like this one.

The probability of at least one

"At least one" questions look hard but have a neat trick. The opposite of "at least one six" is "no sixes at all", which is easy to work out. So find that and subtract it from 1.

In three rolls of a fair dice, P(no six) = 5/6 × 5/6 × 5/6 = 125/216. So P(at least one six) = 1 − 125/216 = 91/216, which is about 0.421.

Here x is the number of rolls and y is the probability of rolling at least one six. It climbs quickly and then levels off towards 1 without ever reaching it. It first passes a half at 4 rolls, when it reaches about 0.518, so with 4 rolls you are slightly more likely than not to see a six.

Experimental probability

Some things can't be worked out by counting outcomes, like whether a drawing pin lands point up. Then you run an experiment and use the relative frequency:

If a pin lands point up 37 times in 120 throws, the relative frequency is 37/120, about 0.308. More trials give a more reliable estimate. You can also go the other way and predict how often something will happen, the expected frequency: multiply the probability by the number of trials. In 600 throws you would expect about 0.308 × 600 ≈ 185 point up landings. Watch the estimate settle down as trials pile up in the course's coin flip simulator.

Beyond the basics

When the same trial repeats a fixed number of times, such as the number of sixes in 5 rolls, the probabilities follow a binomial distribution. The chance of exactly 2 sixes in 5 rolls works out at about 0.161. Our Probability Calculator draws binomial and normal distributions and shades the probability you want, and the Permutations and Combinations Calculator counts outcomes when there are too many to list.

Common mistakes

• Assuming outcomes are equally likely when they aren't. A spinner with unequal sections, or a biased coin, needs the actual probabilities.
• Adding when you should multiply. "And" multiplies; "or" adds.
• Double counting with "or". If the events overlap, subtract P(A and B).
• Forgetting to change the numbers without replacement. After one sock is taken, there are 9 left, not 10.
• Writing probabilities as ratios or odds. Use a fraction, decimal or percentage.
• The gambler's fallacy. After five heads in a row, a fair coin is still 1/2 for heads. Coins have no memory.

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