One sort of probability question asks what the chances are that you have a winning lottery ticket. What’s more likely is that you don’t have a full-fledged winner. Your ticket probably matches only some of the numbers selected. A question that naturally arises from this is, “What is the probability of almost winning the lottery?”
We begin with an understanding of how the lottery works and what assumptions we will make. A participant in a lottery chooses numbers from a specified range, and pays for a ticket with these numbers (sometimes a computer randomly generates the numbers on the ticket). The organization running the lottery randomly generates numbers from this same range. The grand prize is awarded for ticket that matches all of the numbers selected.
There are already several assumptions that we make. The main ones are that the selection of each number is independent, the numbers are chosen without replacement, the order of how the numbers are selected is not important and we are working with a uniform sample space in which every number is equally likely to be chosen.
To calculate the probabilities, we need to know exactly how many numbers are selected, and what range of numbers we are making our selections from. One state lottery involves the selection of six numbers from a total of 53. Jackpots for this lottery are typically in the millions of dollars, so it is a representative example. The ideas discussed in what follows can be used for other situations beyond this specific lottery.
For comparison we will calculate the probability of winning this particular lottery. Since the order of the numbers selected is unimportant, we wish to calculate the number of combinations of 53 items selected 6 at a time. This is denoted C (53, 6) = 53!/(6!47!) = 22,957,480.
Only one of these nearly 23 million combinations is the winning one, so the probability of winning is 1/22,957,480, which is about 0.000004%.
Five Correct Numbers
Now we will calculate the probability of almost winning. Almost is an ambiguous word, so we will consider several situations. First we will see what the probability is of matching all but one number.
The total number of combinations is unchanged, and will remain so for all of our situations. The question becomes, how many of the nearly 23 million possible combinations match exactly five of the six selected numbers? There are C (6, 5) = 6 ways to choose the five correct numbers. There are 47 remaining incorrect numbers. Using the multiplication principle, there are 6 x 47 = 282 possible ways to match exactly five of the six lottery numbers. Thus the probability of matching exactly five of the six is 282/22,957,480, which is about 1/81,410 or 0.0012%.
Four Correct Numbers
Another situation that could be considered almost winning the lottery is to match four numbers, only missing two of them. The probability of this is calculated in a similar way as above. There are C (6, 4) = 15 ways to select exactly four correct numbers. There are C (47, 2) = 1081 ways to choose two incorrect numbers. This means there are 15 x 1081 = 16,215 ways to choose exactly four correct lottery numbers.
There are 22,957,480 total ways to choose six numbers from 53 without regard to order. Thus the probability of exactly four correct numbers is 16,215/22,957,480. This is approximately 1/1416 or 0.07%.
Other Correct Numbers
We will stretch the meaning of the word “almost” in what follows. But since we have all of the tools we need, it is straightforward to determine the remaining possibilities.
There are C (6, 3) = 20 ways to choose exactly three correct lottery numbers out of the six correct ones. There are C( 47, 3) = 97,290 ways to choose three incorrect numbers. Thus there are 20 x 97,290 = 1,945,800 lottery tickets with exactly three numbers correct. The probability of this is 1,945,800/22,957,480, or about 8.5%.
There are C(6, 2) x C(47, 4) = tickets with exactly two numbers correct. The probability of matching exactly two numbers is 2,675,475/22,957,480, which is about 11.7%.
There are C(6, 1) x C(47, 5) = 9,203,634 possible tickets with exactly one number correct. The probability of matching only one number is 9,203,634/22,957,480, which is about 40%.
Finally, there are C(47, 6) = 10,737,573 possible tickets with no correct number. The probability of this occurring is about 47%.
From the above analysis we can see several things:
- It is rare to match exactly five numbers, but this is nearly 300 times more likely than winning the lottery.
- Matching exactly four numbers is about 16,000 times more likely than winning the lottery and nearly 60 times more likely than matching exactly five numbers.
- It’s more likely to match at least one number than to not match any at all.