Probability in a Uniform Sample Space

Have you ever wondered what the likelihood of rolling a particular number or set of numbers on a dice would be, or a particular card from a deck would be? We can make it even simpler and say, "What is the likelihood that a dice when rolled will land on the number 6?"

I'm going to review over the basics of probability in our post today. Let's take the last question, what is the likelihood we would get a particular number on a dice roll? Now assuming we are using a regular 6 sided die we see the following possibilities: {1, 2, 3, 4, 5, 6} Those are really the only numbers we can roll since they are the only numbers listed on the die. So, if we roll a 1 we have 1 out of 6 chances of rolling that 1. If we roll a 2- that is still only 1 number out of 6 possible and we still have a 1 out of 6 chance of rolling that number. Same thing with 3, 4, 5, and 6. Since each possibility is equal, we call that a uniform sample space- all possibilities are uniform.

What if we wanted to take it a step further to see what the likelihood is that we could roll an even number? This event would mean we could roll {2, 4, 6} out of the possible total of {1, 2, 3, 4, 5, 6}. Since there are 3 possible outcomes from a total of 6 we would say the ratio or likelihood of rolling an even number would be 3 to 6.

We can use the following formula to help us solve probability with uniform sample spaces.
The probability (P), of an even  (A)- appearing as P(A) has the following ratio:

                          Number of outcomes associated with the event A
       P(A)=             Number of outcomes in the sample space S



or if we incorporate this into our problem above we get:

Probability (P) of rolling an even number (A)=     3 {2,4,6}
                                                                       6 {1,2,3,4,5,6} 

You can break this down and get 1 over 2... which would also mean 1/2 (or 50%) of the time we could roll and even number and the other 1/2 an odd number. The following video is included to reinforce this idea. Remember, this is probability at it's basics. If we are working with numbers that do not have equally likely outcomes we use a different formula- I will try to discuss this idea in a later post.