What are the chances?

Many of us dream about how awesome it would be to win the lottery. We daydream about the trips we would take, the cars we would buy, never having to work again and spending our days on a sunny beach with a drink in our hand. We have seen the extremely lucky people who have won a lottery on television and cant help but to ask ourselves, realistically, what are the chances?

Probability is a relatively new field in mathematics and was developed to make sense of gambling games and make informed decisions about risk. Since it’s early days, it has grown into a field that beautifully merges logic, calculus, and a little bit of common sense to be one of the most sought after skills in our very predictable times. Before we consider becoming overnight millionaires, let us take a very brief look at some basic statistics. Like every new math that you look at, it is based on logic.

A statement is a declarative sentence that can either be shown to be “true” or “false”. Something either “is” or “is not”. “A square has 4 sides”, “A coin has two sides”, and “3 is an even number” are all examples of statements, albeit the last one is not true. Regardless of whether it is true or false, a statement has the property that it can be “either, or”, never both. Some examples of non-statements are “She is pretty”, “Relax!”, or “What is your name?”. These cannot be proven to be either true or false. We will focus on statements because they are used to predict outcomes. When you flip a coin, it will either be heads or tails, never both, never none. When you roll a six-sided die it will land with a number from 1 to 6 facing up, never none, never more than one number at a time. A simple definition of probability stated mathematically is the ratio

# of desired observations / # of total possible outcomes

This gives us a percentage that we can use to gauge how sure we are about something. The # of total possible outcomes is called the Sample Space. The first thing we do when trying to measure probability is to define our sample space. There are three axioms of probability that, if we think about, pretty much make perfect sense. The first axiom is that the probability of anything happening is between 0 and 100 percent. In other word there is no negative probability and no probability greater than 100%. The second axiom is that the probability that an outcome is in the sample space is 100%. This means that every possible outcome is listed in the sample space and anything not listed will never occur. The third and final axiom is that the sum of all mutually exclusive events listed in the sample space is 100%. Mutually exclusive means that there is no overlap in outcomes, or two things happening at the same time. This means that if you add up all of the probabilities for all of the possible outcomes, they should total 100 percent.

Image: Public domain, via Wikimedia Commons.

Image: Public domain, via Wikimedia Commons.

I like to think of probability like cream cheese. If I were to give you one pound of cream cheese, you can smear as much as you want on however many bagels you want. The cream cheese (probability) will be spread on bagels and any bagels with cream cheese will represent our sample space because they have a non-negative probability assigned to them. Any bagels without cream cheese will not be in our sample space because they have no probability assigned to them and thus are considered impossible outcomes. Let’s suppose that you chose to smear the one-pound of cream cheese on half a dozen bagels. You did not smear the cream cheese evenly and that is OK because in real life some outcomes are more likely than others. Here is what the bagels look like:

Bagel 1: 1/6 lb

Bagel 2: 1/12 lb

Bagel 3: 1/8 lb

Bagel 4: 1/3 lb

Bagel 5: 1/4 lb

Bagel 6: 1/24 lb

As long as you use all the cream cheese, we can apply our axioms,

Axiom 1: I can say that the amount of cream cheese you smeared on an individual bagel is a non-negative amount not exceeding one pound.

Axiom 2: If I were to pick a random bagel and it had cream cheese, it would have to be one of these six bagels listed above.

Axiom 3: If I were to scrape all the cream cheese you smeared off the bagels, it would weigh one pound.

Using this as our first example, the possible outcomes of selecting a bagel are the numbers 1 through 6, or stated mathematically as a set, {1,2,3,4,5,6}. Let us define the event A, as “selecting an even numbered bagel”. We first look at how many events in the sample space satisfy this condition, and then sum their probabilities. There are three even numbers in our sample space, namely {2,4,6}. The probability of selecting an even numbered bagel is sum of the amount of cream cheese on these bagels. So we say that the probability of A, written

P(A) =(1/2)+(1/3)+(1/24)=7/8

Let’s define another event, B, as “selecting a bagel numbered greater than 3”. The events in our sample space are the bagels {4,5,6}. Summing up their cream cheesiness, we obtain

P(B)=(1/3)+(1/4)+(1/24) =5/8

As you can see, the probability of selecting an even numbered bagel is greater than the probability of selecting a bagel numbered 4-6.This is a very brief and basic example of probability. There are many technicalities that have been breezed over and probability can become extremely complex and sophisticated, involving counting techniques such as combinations or permutations in order to account for very large numbers.

One final concept needed in order to discuss our lottery example is the Expected Value. The expected value, surprisingly, is what you should expect if you were to perform the trials for long periods of time. By “values” we mean what events we are considering, in our case, the whole numbers 1-6. The formula for the expected value, denoted E[x], is the sum of all x*p(x), where x is the values and p(x) is the probability of each respective value. What this says in English is that the expected value is the sum of products of all the values and their respective probabilities. Let’s cement this idea with an example with our die but first a question. If a suspicious looking man wearing a trench coat approached you and held a fair die in his hand and told you that he would pay you the equivalent of whatever number you rolled on the die, and the cost of rolling the die was $2, should you statistically do it? Lets calculate the expected value of our rolls. Once again, assuming the die is fair, the probability of rolling any number 1 thru 6 is 1/6, because we have 6 possible outcomes. Invoking our expected value formula, let’s multiply our values (1,2,…,6) with their individual probabilities, 1/6, and add everything up and see what we should expect.

1(1/6)+2(1/6)+3(1/6)+4(1/6)+5(1/6)+6(1/6)=

1/6+2/6+3/6+4/6+5/6+6/6= 21/6= 3.5

According to our Expected Value calculations, we should expect $3.50 from this challenge. So is it worth the $2? Absolutely! The expected value computation means that if you kept playing, say, 100 times, you should expect to make $350, so if you paid $200 for it, you would still make a profit.

 

In Charles Wheelan’s book Naked Statistics, he uses the Expected Value formula to guesstimate the chances of winning the Illinois Dugout Doubler. On the back of most lottery tickets there is really fine print giving the probabilities of winning every prize, not just the jackpot. We will use the Illinois Dugout Doubler as an example, noting that each ticket costs $1. The possible prizes are $2, $4, $5, $10, $25, $50, $100, $200, $500, and $1,000. Their respective probabilities are listed along with them. Lets calculate the expected value for our hard earned $1 that we are considering spending.

 

E[x]= 1/15($2)+1/43($4)+1/75($5)+1/200($10)+1/300($25)+1/1,589($50)+1/80,000($100)+1/16,000($200)+1/48,000($500)+1/40,000($1000)=

 

$.13+$.09+$.07+$.05+$.08+$.03+$.01+$.01+$.01+$.03 = $.51

 

So the expected value of our $1 ticket is 52 cents. This is equivalent to saying that if you were to buy one thousand tickets for $1 each ($1,000 spent) after all the wins and losses, because you will win, you should expected to end up with $520 in wins. Not a very statistically sound way to invest your money. But then again, lady luck smiles at people from time to time and probability is just a way to measure and gauge, it is not definite. Good luck!

 

References

Wheelan, Charles. Naked Statistics. New York: W.W. Norton. Print.

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