Ever wondered what the odds are of winning a prize in the UK National Lottery? Or how likely you are to win the jackpot? Well wonder no more - this article both answers these questions, and shows you how to do the calculations yourself.
Please note: This article relates to the old format of the UK National Lottery which involved picking six numbers from 1 to 49. This has since change to 1 to 59. We've left this here for posterity but will be writing an update version shortly.
What Are The Odds Of Winning The Jackpot?
To win the jackpot in the National Lottery (Lotto) you must match all six winning numbers to the numbers on your ticket. We can work out the chance of this happening as follows.
For the first winning number there are 49 balls that can be drawn, and each has an equal chance of being selected. On your lottery ticket you have selected 6 numbers - the first ball drawn just has to match any of your chosen numbers as the order does not matter. The chance of the first ball drawn matching any number on your ticket is therefore 49 divided by 6.
For the second winning number, there are now 48 balls to draw from and you have 5 remaining unmatched numbers on your ticket. The chance of the second ball to be drawn matching your ticket is therefore 48/5. For the third winning number you have 47 balls and 4 selections, and so on up to the sixth winning number when there are 44 remaining balls and only one unmatched spot on your ticket.
Multiplying all of these numbers together gives us:
49/6 x 48/5 x 47/4 x 46/3 x 45/2 x 44/1 = 13,983,816 (approximately 14 million)
So the chance of winning the UK National Lottery (Lotto) Jackpot is approximately 1 in 14 million.
What Are The Odds Of Winning A Specific Prize?
You do not have to win the jackpot to receive a prize - matching 3 numbers or more would result in a win. The following are the odds of winning a specific prize in the UK National Lottery:
This is the chance of matching exactly 3 numbers, as matching more would result in a higher prize. To calculate the odds we first work out the following:
- The chance of matching the first number drawn: 49/6 (49 possibilities, divided by 6 chances)
- The chance of matching the second number drawn: 48/5 (48 possibilities, divided by 5 chances)
- The chance of matching the third number drawn: 47/4 (47 possibilities, divided by 4 chances)
- The chance of not matching the fourth number drawn: 46/43 (46 possibilities, divided by the remaining 43 balls that do not match your selection)
- The chance of not matching the fifth number drawn: 45/42 (45 possibilities, divided by the remaining 42 balls that do not match your selection)
- The chance of not matching the sixth number drawn: 44/41 (44 possibilities, divided by the remaining 41 balls that do not match your selection)
If we multiply these together we get:
49/6 x 48/5 x 47/4 x 46/43 x 45/42 x 44/41 = 1133.119
There are actually 20 ways of matching exactly 3 balls as you could match the first second and third balls as in the example above, or you could match the fourth fifth and sixth balls, or any combination in between.
So the odds of matching exactly three numbers are:
1133.119/20 (chance of matching three balls divided by the number of possible combinations) = 56.65593
Therefore your odds of winning a prize for matching three numbers is approximately 1 in 56.7.
Calculating the odds of matching 4 numbers is similar to the way in which we calculated the odds of matching 3 numbers, except that this time we first take the odds of matching the first 4 numbers and the odds of not matching the remaining two numbers:
49/6 x 48/5 x 47/4 x 46/3 x 45/43 x 44/42 = 15,485.95
We now divide this by the number of possible combinations (15) to get:
15,485.95/15 = 1032.397
Making your odds of winning a prize for matching four numbers approximately 1 in 1032
The chances of matching exactly five numbers is slightly different, because at this point you need to take into account the bonus - if you were to match the bonus ball you would win a bigger win. To do this we simply multiply the chance of matching five numbers by the chance of not matching the final number or the bonus ball (43/42 - 43 remaining balls divided by the remaining 42 balls that do not match your selection):
49/6 x 48/5 x 47/4 x 46/3 x 45/2 x 44/43 x 43/42 = 332948
Divided by the number of possible combinations (6):
332948/6 = 55491.33
Making your odds of winning a prize for matching 5 numbers approximately 1 in 55,491.
Five Numbers + Bonus Ball:
To calculate the chance of matching five numbers plus the bonus ball, we can use the calculation for matching five numbers and change the bonus ball part. Now we are looking at the chance of matching five numbers multiplied by the chance of not matching the final number multiplied by the chance of matching the bonus ball (43/1 - 43 remaining balls and one remaining chance):
49/6 x 48/5 x 47/4 x 46/3 x 45/2 x 44/43 x 43/1 = 13,983,816
Divided by the number of possible combinations (6):
13,983,816/6 = 2,330,636
Making your odds of winning a prize for matching 5 numbers plus the bonus ball 1 in 2,330,636.
The Jackpot - 6 Numbers:
As shown above, this is 1 in 13,983,816
So What Are The Odds Of Winning Any Prize?
With these numbers we can also calculate the chances of winning any prize in the National Lottery - approximately 1 in 53.7.