I don’t buy lottery tickets for the simple reason that in most cases the house (in this case the government) offers terrible odds by keeping too much of the pie for its own greedy little ends. It’s often referred to as a “tax on stupidity” by preying on wish-fulfillment fantasies or on people that are in dire need of cash.
This weekend however, odds are that I’ll buy two tickets.
Here’s how the numbers break down (and thanks to the secret statistician that helped me with the numbers):
There are 49 possible numbers, and you must pick a set of 6 numbers from those 49 that exactly match what the lottery randomly selects to win the jackpot.
To get a unique sequence of 6 numbers you have 49 choices for the 1st, 48 choices for the second, etc. Therefore there are 49x48x47x46x45x44 unique sequences, or 10,068,347,520 (that’s about 10 billion) unique sequences of 6 numbers out of 49.
The lottery however is not based on unique sequences, otherwise you would not only be picking the numbers but the order they came out of the Ryo-Catteau Tulipe ball machine that the lottery uses to randomly select the numbers. Put another way, there is no difference between the set (5, 10, 15, 20, 25, 30) and the set (30, 25, 20, 15, 10, 5).
How many unique sequences, or permutations are there in a set? There are 6 possibilities for the 1st number in any set, 5 possibilities for the 2nd, and so on. Therefore there are 6x5x4x3x2x1 sequences in each set, or 720 sequences per set.
If we divide the 10,068,347,520 total unique sequences by the 720 unique sequences for each set we get the odds of 1 in 13,983,816 (that’s about 14 million). Those odds are approximately the same as flipping a fair coin and having it come out heads 24 times in a row (2^24 = 16,777,216).
We can use these numbers to calculate the “expected value” of each ticket by taking the estimated size of the jackpot (about $55 million) and dividing it by the odds (about 14 million). This results in an expected value of $3.93 per combination set. The price of a ticket is $2.00 - so the odds are in your “favour” because you are paying only $2.00 for a ticket combination that is statistically worth almost $4.00.
But there’s a catch. If there is more than one winner, the prize is shared by all of the holders of that particular combination set. If there are two winners for example, each person will receive half of the pot, or about $27.5 million. The expected value per combination set in this scenario is now only $1.96 per ticket. This is less than the $2.00 price of a ticket, and so the expected value of your ticket is less than the price: a losing proposition.
One way around this is to buy multiple tickets WITH THE SAME NUMBER COMBINATION. In this way, if there is one other winner, you will hold two winning tickets to her one, and thereby receive two thirds or 67% of the pot. 67% of $55 million works out to about $37 million. This works out to an expected value of $2.63 per combination set and $5.26 for the two ticket set vs. a price of only $4.00 for the combo.
Bang, the odds are in your favour, and statistically speaking, this is a better wager than buying two tickets with different combinations.
So that’s what I’m going to do. Wish me luck :-)