How to find a probability, from coin flips to cards

What probability measures

A probability is a number from 0 to 1. An impossible event, such as rolling a 7 on a standard die, has probability 0. A certain event, such as rolling a number less than 7, has probability 1. Everything else falls in between, and an event with probability 1/2 is as likely to happen as not.

Probabilities can be written as fractions, decimals or percentages: 1/4, 0.25 and 25% all say the same thing. Fractions are usually easiest while you calculate, because they keep values exact.

A probability describes the long run, not the next try. A fair coin has probability 1/2 of landing heads, which means that over many flips about half will be heads. It does not mean that two flips will give one head and one tail.

The basic formula and the complement

List or count the sample space, which is every possible outcome. If the outcomes are equally likely, then P(event) = (number of favourable outcomes) ÷ (total number of outcomes). The word favourable just means the outcomes you are asking about.

The complement of an event is everything else: the event not happening. Since one or the other must happen, P(not A) = 1 − P(A). If the chance of rain is 0.3, the chance of no rain is 0.7.

The complement is often the quickest route to an answer that contains at least. The chance of at least one six in two rolls of a die is 1 minus the chance of no sixes: 1 − (5/6 × 5/6) = 1 − 25/36 = 11/36, a bit under one third.

Worked example: rolling two dice

Two dice give 6 × 6 = 36 outcomes, each equally likely. Listing them as ordered pairs, first die then second, avoids the common mistake of treating a total of 7 as one outcome among only eleven possible totals.

Or: adding probabilities

For A or B, add the probabilities, then subtract the overlap so it is not counted twice: P(A or B) = P(A) + P(B) − P(A and B). If the two events cannot happen together, they are called mutually exclusive, the overlap is 0, and you simply add.

And: independent and dependent events

For A and B, multiply. Two events are independent when one does not change the chance of the other, like two coin flips: P(heads, then heads) = 1/2 × 1/2 = 1/4. Events are dependent when the first changes what is left for the second, as when you draw without putting the first item back. Then the second probability must be worked out after the first has happened.

Theoretical and experimental probability

Theoretical probability comes from reasoning about equally likely outcomes, as in the examples above. Experimental probability comes from doing the thing and counting: if a spinner lands on red 27 times in 100 spins, the experimental probability of red is 27/100.

The two rarely match exactly over a few trials, and that is expected. As the number of trials grows, the experimental probability tends to settle closer to the theoretical one. This is called the law of large numbers, and it is why a few unusual results are not evidence that a coin or die is unfair.

Common mistakes and misconceptions

Probability is one of the topics where intuition most often misleads. Watch for these:

• The gambler's fallacy: after five heads in a row, tails is not due. Each flip of a fair coin is still 1/2.
• Treating outcomes as equally likely when they are not. The eleven totals from 2 to 12 with two dice are not equally likely.
• Adding when you should multiply. Or usually means add (minus any overlap); and usually means multiply.
• Double counting the overlap in an or question, such as counting the king of hearts twice.
• Forgetting to change the numbers for the second draw when there is no replacement.
• Getting an answer above 1 or below 0. A probability can never be outside that range, so such an answer signals an error.

How to practise and check your work

Tree diagrams and tables make multi-step problems visual. Draw a branch for each outcome, write the probability on it, multiply along a path for and, and add the paths that fit the question. A two way table does the same job for two dice or two categories.

Two checks catch most errors: every probability must lie between 0 and 1, and the probabilities of all possible outcomes must add up to 1. For a quick experiment at home, flip a coin or roll dice 50 times, record the results, and compare them with the theory. In Common Core math these ideas start in grade 7 and continue in high school with conditional probability.