Sample space

In Probability, all possible outcomes of an action, like picking a card from a deck of cards, is called the sample space . So, in our example of picking a card from a pack of cards, the sample space is made up of all of the cards in the pack.

If a regular six-sided dice is rolled then the sample space for the number it lands on is 1, 2, 3, 4, 5, 6.

The sample space is some\times written as the symbol S and the members of S are often written in a list in curly brackets, like this:

S = {1, 2, 3, 4, 5, 6}

If a coin was tossed, it could land on heads (H) or tails (T). The sample space for this is S = {H, T}.

If two coins were tossed at the same time then S = {HH, HT, TH, TT]

A sample space can also be shown by a diagram. If two regular six-sided dice are rolled and the numbers that the two dice land on are added together then we can show the sample space like this:

The sample space is made up of the numbers highlighted in blue. Each square, or event, in the sample space is equally likely. There are 36 (6 x 6) highlighted squares in the sample space diagram. We can use the sample space diagram to work out the probability of the numbers on the two dice adding up to different amounts.

For equally likely events, like in the sample space here:

Let’s work out the probability of getting 2

What P(7)?

What is P(not getting a 7)? We can write this as P(7′)

We can count the number of ways of not getting a 7 in the sample space. There are 30 ways of not getting a 7 and so

Notice that

This makes, sense, right? The probability of something happening + the probability of it not happening is 1. It is certain.

We can write this is mathematical language. Let A represent an event then

We can rearrange this to get

So we could have worked out P(7′) from P(7)