EMAT 6700
Probability
We have all heard
of the Georgia Lottery. Some of us even went to college from that money. Maybe
you have played the lottery, maybe you have not played, but we all know what
the lottery is and that some people win lots of money. The question I want
to propose to you is "What are the chances of you, a single winner, reciving
the big prize?" We are going to look at a few concepts that help us to
understand questions like this one.
You might ask how in the world can I even begin to think about something like that? Well that is what I am here for. First we need a problem so solve. I did a little research about the Georgia lottery. We first need to understand how play the lottery. I chose the Big game as the one to play, since it involves the most money and the smallest chance to win. They way one plays this game is any time during the week go to any store that sells lottery tickets and pick out your numbers. In the Big Game, one selects 5 numbers from 1 to 50 and then one other number from 1 to 36. You are probably worried about that last number. Don't worry about that now we will discuss that a little later. Let's first consider the chances of the first 5 numbers.
Given that there are 5 numbers that are "special" or the chosen numbers and I have to pick them from 50, the chance that I have to pick the first one is 5 out 50. We want to find the best chances that we have so let's assume that we are really luck and that each time we choose we pick one of those numbers. If we didn't pick one each time then we will not win anyway. So let's continue. When we choose the next number we only have 49 numbers to choose from, and only 4 of the winning numbers left. Obviously we have a 4 out of 49 chances to draw another winning number. This process continues down to the last number where we have 1 chance out of 46 numbers to pick the last winning number.
Now you as well how to we find the probability of choosing all five numbers at the same time. That is much easier that most would think. Since we have already done the hard part we now multiply all the probabilities together.
Now that we have established the chances of picking the first five numbers we need to address the last number. We have only 1 chance to choose 1 number out of 36. This time the calculations are trivial. we have a chance of 1/36 to pick the last number. And to finsh the calculation we need only to multiply the two probabilities together.
So our conclusion is if you play the lottery 76,275,360 times in one week you should win the lottery at least one time.
I will talk a little about combinatorics now. We want a practial way to develop this new concept or counting method. Lets take the lottery example but on a smaller scale. Let's say there are 5 numbers and we can choose 2 to win the big prize. We will say "5 choose 2" to refer to this probability. On a more abstract level we can generalize to n numbers and we choose m out of those n numbers. I want to introduce another term called "factorials" represented by a "!." This is a function that multiplies all the numbers smaller that the chosen number. For example 5! = 5*4*3*2*1 = 120.
Let's consider how we would count up the different ways to win this smaller lottery. If you pick one number then you can pick 5 different numbers. So you can pick 1 number 5 times. Now let's pick two numbers. If you pick the first one then you have 4 other choices. Or you could pick the second one with 3 other choices (since we don't want to repeat a choice). And all the way down to picking each of the numbers we will have the following chance of picking the two winning numbers.
| First Choice | Second Choice 1 | Second Choice 2 | Second Choice 3 | Second Choice 4 |
| 1 | 2 | 3 | 4 | 5 |
| 2 | 3 | 4 | 5 | |
| 3 | 4 | 5 | ||
| 4 | 5 |
We don't need to choose 5 first since it has been chosen with each number already. This all the possible combinations of choosing 2 numbers out of 5. Mathematically we represent this with the following:
We read this mathematical term 5 choose 2. Or generally n choose m. The n represents the total number of elements and the m represents the total number of choices from n. Here is another example in Micosoft Excel. This time we are working with 10 numbers and choosing 3.
Try figuring out what the chances are when you have 30 numbers to choose from and 4 to pick as your choices. Click here for a blank Excel worksheet.
Try your counting skills at this problem. There are 10 people in a room that do not know each other. The object is to for everyone to shake hands with everyone else only one time and the order does not matter.