Calculating Probabilities

MathsDirect

Before we look at calculating probabilities, there are some definitions that you need to know. These should be familiar to you from GCSE ( and before ), so they only need to be quickly mentioned.

If you are looking at the probability of rolling a 6 on a regular die, then each roll is called a trial.

Clearly there are 6 possible results from rolling a die. Each of these is referred to as an event.

When you combine all the possible events, you get the sample space ( also sometimes referred to as the possibility space).

Theoretical Probability

You will already be familiar with the method of calculating probability.

Probability =

No of Good Events

No of Possible Events

where a good event is the result that you are looking for ( a six )

The number of possible events is just the number of events in the sample space.

You get a clearer idea of how this definition works, if you look at the case of rolling 2 dice and scoring 7.

This diagram shows the sample space for rolling 2 dice.

The numbers in the grey boxes are the scores of the individual dice. The rest are the combined scores (2 & 3 gives 5)

	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

You can see that there are 36 events in the sample space. 6 of these events give you a total of 7.

The probability of scoring 7 is therefore

Using the same sample space, we can find the probability of scoring 4. You can see that in this case there are 3 good events, so

We could also work out the probability of scoring more than 10. We can now have either 11 or 12. When more than one outcome will do, it is called a compound event.

There are 3 ways of scoring either 11 or 12, so

Experimental Probability ( Relative Frequency )

The method of theoretically determining probability, described above, depends upon two things

1	Being able to define all the possible outcomes. That is being able to fully define your sample space
2	All the possible events must be equally likely. That is your die must be unbiased.

Practically, it is unlikely that the above conditions will be satisfied as soon as you stop dealing in dice and packs of cards. The real world is not so easy to define and things tend not to be equal. You therefore need some method of determining probabilities experimentally.

The formula for experimental probability is essentially the same as the theoretical formula, but is based upon you actually conducting trials.

Probability =

No of Good Results

No of Trials

where a good result is rolling a six and the number of trials is how many times you roll the die.

Samples

Suppose you wanted to find the probability that a random student in a school watches Eastenders.

You could ask every student in the school and divide the number that said they watched it, by the total number of students. This would probably, however, involve asking around 1000 students, which would be extremely tedious. Instead, you would probably choose a sample, of say 50 students, which would represent the whole school population.

It is important how you choose the 50 students. They should be randomly selected, to avoid attaching too much importance to, say sixth formers. A random selection would be expected to be fairly evenly spread over the different year groups.

Suppose from your sample, 30 students watched Eastenders. Then you could say that the probability that a student watches Eastenders is

When you determine a probability this way, it is also referred to as a relative frequency.

You could use this figure to predict how many students in total watch Eastenders. Such a result is called the expectation.

where n is the number of students. So if we had 1000 students, we would expect 600 to watch Eastenders.

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