What a lottery draw asks of you
Many lotteries ask a player to pick a small set of numbers from a fixed range, then draw the same quantity of numbers from that range without putting any back. Real games differ in how many numbers are picked, how large the range is, and whether there are bonus balls or a second pool of numbers. This article uses one illustrative format, picking 6 numbers from 1 to 49, because the arithmetic is clean and the same method applies to other formats. For any real game, the published rules decide the actual figures.
In this format the order of the drawn numbers does not matter. A ticket with 3, 11, 19, 27, 38, 46 wins the jackpot whether those numbers come out in that order or any other.
Counting the possible tickets
To find the chance of matching all six numbers, count how many different tickets exist and note that exactly one of them is the winner.
Imagine drawing the numbers one at a time. There are 49 choices for the first number, 48 for the second, 47 for the third, then 46, 45 and 44. Multiplying gives the number of ordered sequences:
49 × 48 × 47 × 46 × 45 × 44 = 10,068,347,520
That counts the same set of six numbers many times, once for every order in which it could have appeared. Six numbers can be arranged in 6 × 5 × 4 × 3 × 2 × 1 = 720 ways, a product known as 6 factorial. Dividing the ordered count by 720 removes the repeats:
10,068,347,520 ÷ 720 = 13,983,816
So there are 13,983,816 possible tickets, each equally likely to be drawn. The jackpot chance for a single ticket is 1 in 13,983,816, or roughly 1 in 14 million. Written as a percentage that is about 0.000007 per cent. This kind of counting is the subject of combinations and permutations, and the order-ignoring version used here is called a combination.
The same answer as a chain of shrinking chances
There is another way to see it. The chance that the first drawn number is one of your six is 6 out of 49. Given that, the chance the second drawn number is one of your remaining five is 5 out of 48, and so on:
6/49 × 5/48 × 4/47 × 3/46 × 2/45 × 1/44 = 720 / 10,068,347,520 = 1 / 13,983,816
Each factor is a modest-looking fraction, but six of them multiplied together shrink very quickly. That is the core reason the odds are so long: the jackpot needs several separate things to go right, and the chances multiply.
The smaller prizes
Many lotteries pay smaller prizes for matching fewer numbers. The counts below use the same illustrative 6 from 49 draw and ignore bonus balls. To match exactly five, you need five of the six winning numbers (6 ways to choose which five) and one of the 43 numbers that were not drawn (43 ways), giving 6 × 43 = 258 winning tickets.
| Numbers matched | Winning tickets out of 13,983,816 | Approximate odds |
|---|---|---|
| 6 | 1 | 1 in 13,983,816 |
| 5 | 258 | about 1 in 54,201 |
| 4 | 13,545 | about 1 in 1,032 |
| 3 | 246,820 | about 1 in 57 |
The pattern is typical. The lowest prize tier is within reach of a few dozen tickets, while each step up multiplies the difficulty by a large factor. The gaps between rows are far bigger than the gaps between the match counts suggest, which is why people often underestimate how steeply the odds climb.
Why adding numbers makes it so much worse
Widening the range has a dramatic effect. Picking 6 numbers from 59 instead of 49 gives 45,057,474 possible tickets, about 3.2 times as many, from adding only ten numbers to the pool. Adding a required bonus number from a second pool multiplies the total again. This is why games that advertise very large jackpots usually have very long odds: a large prize and a tiny chance tend to come as a package.
What the odds do and do not tell you
The figure of 1 in 14 million describes one ticket in one draw. It does not mean a win will arrive after 14 million tickets, just as flipping a coin twice does not guarantee a head. Over a large number of tickets the chance builds up gradually, and the outcome is still uncertain.
Each draw is also independent of the last. Machines and number generators have no memory, so a number that has not appeared for a long time is not “due”. Believing otherwise is the gambler’s fallacy, and it applies to lotteries as much as to roulette.
Odds and percentages are two ways of describing the same thing, and tiny chances are hard to feel intuitively. Calling a chance “one in a million” does not convey how rare it is, and most people treat all low-probability events as roughly the same size, whether they are one in a thousand or one in ten million.
Finally, long odds are only half the story of what a ticket is worth. Prizes are typically set so that the average return is below the ticket price, which the article on lottery ticket value works through with numbers. Playing for entertainment with money you can afford to lose is a personal choice, but no pattern, system or lucky selection changes the odds. Gambling can become harmful for some people, and anyone who is struggling to stay within limits should consider contacting a local support service, which can offer confidential help. The topic of problem gambling is covered separately on this site.