Sunday, February 15, 2009

Defining and Calculating Probability

Yesterday we looked at sample spaces, outcomes, and events. Today we will look at ways to determine probability.

Again, consider the die. Imagine we wanted to know the probability of rolling an odd number. The sample space for a die is [1,2,3,4,5,6] and the outcomes of rolling an odd number are [1,3,5] so to calculate this probability we take the number of odd outcomes and divide it by the total number of outcomes.

In this case it is 3/6 or 1/2 or 50%. This is only true if the die is not slanted, that is to say, all outcomes are equally likely.

The second way of calculating probability can be used to test if the die is honest. This is found by rolling the die many times, around 100 and seeing how many times an odd number comes up. So in this case we calculate the probability by counting the number of times an odd number occurs over the number of times we roll the die.

Suppose we do this 100 times and find we rolled 55 odd numbers. Then we see that 55/100 times we rolled an odd, or 55% of the time. This is close enough to 50% to say that the die is honest. After rolling the die 1000 times or even 10,000 times we would expect the number to get closer and closer to 50%.

This second method of probability can also be used in times we we do not know the sample space. Let us say we want to know the probability of making a sale when a customer walks into our shop. There is no way to calculate this theoretically(the first method) and we must do it by counting. Assume that 100 people walk into the shop, and only 20 people end up making a purchase, then we conclude that the probability of making a sale is 20/100 or 20%.


The Jamaican said...

Is there no theoretical way of calculating the first method?

Paul House said...

The first method is the definition of probability.

The sample space is defined.
The number of outcomes have to be counted in that sample space and then divided by the sample space to get the probability of a particular event happening.

That is the theory. It is the theoretical way.