Here is a nice video by Salman Kahn, the founder of Kahn's Academy, that shows how you calculate the probability of winning the Mega Millions jackpot. Probability is just a fancy word for chance.
If you buy one lottery ticket your chance or probability of winning is 1 in 175,711,536! That's pretty slim if you ask me!!! That's why I haven't bought a lottery ticket until last night; one of my buddies on Facebook convinced me to.
So I got thinking about this a little more. Because I know a little probability theory, I wanted to play with this a little. Okay, scratch that. I probably know more probability theory than most people. I am just being modest :-)
Since I did buy one lottery ticket I wanted to know what my expected value is. In other words I want to know the amount I can expect to win (or lose) by playing the Mega Millions lotto one time. I am assuming that I and everyone else playing can win or lose. By this I mean two things:
1.) If all the numbers on my ticket match the winning combination I win the big jackpot. If they don't match I win nothing. Even if 5 out of 6 of the numbers on my ticket match the winning combination I still win nothing (I don't think this is the case in reality).
2.) Only one person can win the jackpot. In other words no more than one person can have a winning ticket. If I buy more than one ticket I cannot have more than one winning ticket.
Now let me explain the concept of expected value. If all the numbers on your ticket match the winning combination you expect to be $540 million richer. If not, you don't expect to win anything. Simple, right? Not so fast.
The expected value of playing the Mega Millions Lottery is just the weighted average of all possible outcomes. As I have outlined it in this blog I am only allowing two possible outcomes for playing the Mega Millions lotto; winning the big jackpot or winning nothing at all.
I am going to label the two outcomes with a capital letter for convenience.
W: winning the big jackpot
L: not winning anything ('L' for losing)
I said the expected value is just the weighted average of all possible outcomes. This just means I multiply the value of each outcome by its corresponding probability and sum them up. If all outcomes are equally likely (i.e. each outcome has the same probability) I just add the values of all possible outcomes together and divide the total by the number of outcomes. If there were 5 possible outcomes with the same probability, I just sum up the values for the 5 outcomes and divide by 5. This is how you normally calculate the average.
Whenever the outcomes are not equally likely (i.e. at least one of the outcomes has a probability different from the others) I instead calculate the weighted average of all those outcomes. This will give me the expected value.
To see how to do this let's make a table for the two outcomes in question.
W | 0.00000000569 |
L | 0.99999999431 |
This table just lists all possible outcomes for the Mega Millions lotto and their corresponding probabilities. Notice the probabilities are not the same, which is why I calculate the weighted average.
Remember, a probability is just the number of ways that a particular outcome can occur divided by the total number of ways for all possible outcomes (as explained in the video). Whenever the probability of an event is 0 that means the event has no chance of occurring. If the probability of an event is 1 the event has every chance of occurring (i.e. the event will happen). The closer the probability of an event is to zero the less likely that event is going to occur. Likewise, the closer the probability of an event is to 1 the more likely it is going to occur. Probabilities are usually expressed as decimals or fractions and they can never be negative or bigger than one.
Continuing with our example, if I buy only one ticket and it is a winning ticket then the probability of winning is 1/175,711,536 (the slash '/' stands for dividing). If I buy only one ticket and it is not a winning ticket the probability of losing is 175,711,535/175,711,536. You should notice the number 175,711,535 is one less than 175,711,536. The decimals in the table were obtained using a calculator.
In case you missed the video the number 175,711,536 represents the total number of ways of generating the first 5 numbers and the 6th 'mega' number. The 6 numbers on your lotto ticket are generated in a similar fashion. I will not explain how to calculate 175,711,536 in this blog!
The only thing I need now is the value for each outcome. The value represents my net winnings after buying one ticket. This is just the amount of money I win minus the amount of money I spend to buy a single lotto ticket. Here is a table showing each outcome, its value and its corresponding probability.
W | 539,999,999 | 0.00000000569 |
L | -1 | 0.99999999431 |
If I win I win $540 million minus $1. This is because I spend $1 to buy the ticket so my net winnings are $539,999,999. If I lose I win $0 minus $1. For the same reason my net winnings are -$1. I use the term net winnings loosely here. Whenever my net winnings are negative I am really losing something. Confusing, huh?
Therefore, my expected value is going to be $2.07. That's right, I can expect to win $2.07 by playing the Mega Millions lotto. I emphasize expect, because that does not mean I will actually win $2.07. The expected value does not accurately reflect what I will win if I buy a single lotto ticket. It tells me how much on average I can win if I play one ticket. In reality (as I have outlined it in this blog) I either walk away with the jackpot or I walk away with nothing, minus the $1 I spend on the ticket.
The message to take home from this blog is that you should play the Mega Millions lotto and buy one ticket. Why? Because the expected value of buying one ticket is positive. This means the lotto is in your favor. So, please go out and buy a ticket before tonight. You have nothing to lose!
Thank you for reading my blog! I will create a second blog shortly where I will explain what happens to the expected value when you buy more than one ticket.