A total of 70 balls went into a drum for a lottery drawing. Five were pulled out: 56, 66, 67, 68 and 69.
Maybe you noticed there are four consecutive numbers within the five. Maybe you wondered how rare that is. And then, maybe, you said to yourself, “Ah, that sounds like hard math, and I barely remember what an exponent is. I guess I’ll never know.”
The good news is we’ve got the answer.
The drawing in this case was for Tuesday’s multistate Mega Millions jackpot, which had no winner and pushed the prize to an estimated $382 million cash payout.
The four-in-a-row pattern is certainly not typical in recent winning Mega Millions numbers. With two drawings a week, there have been several sets of two consecutive numbers in the last year, and in September there were three in a row, 55-56-57, but no drawings with four consecutive numbers.
So the chances must be a bazillion-to-one or something, right?
Let’s find out. First off, we must determine how many possible combinations of the five numbers there are. The formula, for the math savvy, is n!/(r!(n-r)!), where “n” is the total number of balls (70), and “r” is the number of balls drawn (five).
(For those whose memories of high school math are nightmarish, the exclamation point means “factorial,” the number multiplied by each integer below it. So 3! = 3×2×1 = 6, and 5! = 5×4×3×2×1 = 120.)
That calculation gives you about 12 million possible combinations. But how many of them have four consecutive numbers, such as 66-67-68-69?
(The real math starts here. Skip the next three paragraphs, if desired.)
Well, there are 67 different runs of four numbers, from 1-2-3-4 up through 67-68-69-70. In a full lottery drawing, each of those runs would have one additional number drawn, and with four numbers taken out of 70, there are 66 possibilities for that extra number.
And 67 × 66 would seem to make 4,422 combinations that have a run of four (or more). However this double-counts the runs of five — for example, the run 2-3-4-5-6 is counted twice: once as 3-4-5-6 plus a 2 and once as 2-3-4-5 plus a 6. That reduces the 4,422 combinations by 66, to 4,356.
So, divide the 12,103,014 million possible combinations by the 4,356 combos that include a run of four or more and you get a likelihood of 1 in 2,778.
With two drawings a week, that’s roughly every 27 years or so, on average.
So, yes, it’s unusual. But aren’t many combinations that get pulled unusual in one way or another? In this case, there were four numbers in a row, but other interesting number groupings might easily emerge.
“There are many patterns,” said Elchanan Mossel, a math professor at Massachusetts Institute of Technology. “If you find a particular one you may say: ‘Wow! How did it happen?’ But there are other patterns that we would detect with similarly small probabilities: all numbers divisible by three, a run of three and a run of two, etc., etc. So if we search for many patterns and find one, it is not so unlikely.”
And if you think the odds on four numbers in a row are long, consider the chances of actually winning the thing.
First, you have to hit all five numbers exactly. The odds of that are more than 12 million to 1. But then you also have to hit another number (“the Megaball”) exactly, upping the odds to more than 300 million to 1.
Maybe another retirement plan would be wise.
The post A Lottery Drawing Included Four Consecutive Numbers. What Are the Odds? appeared first on New York Times.
