The Chances Of Winning The UK National Lottery
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The Jackpot - 6 Numbers (Typical prize: £2 million)
6 numbers are drawn at random from the set of integers between 1 and 49, which means there are 49!/(6!*(49-6)!) combinations of numbers - this means that the jackpot chance is 1 in 13,983,816 or approximately 1 in 14 million.
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5 Numbers + Bonus Number (Typical prize: £100,000)
You are still matching 6 numbers from the 1 to 49 set as above, but you can now do it in 6 different ways (by dropping each of the main numbers in turn), therefore the chance is 1 in 13,983,816/6, which works out as 1 in 2,330,636.
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5 Numbers (Typical prize: £1,500)
This is 42 times more likely than getting 5 numbers + the bonus number - the chance is 1 in 2,330,636/42, which evaluates to 1 in 55,491.33333.
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4 Numbers (Typical prize: £65)
Firstly, let's take the case of the first 4 of your numbers matching and the last 2 not matching.
In this single case (where each set of chances relies on the previous event occurring):Chance that your 1st number matches a winning lottery number is 1 in 49/6.
Chance that your 2nd number matches a winning lottery number is 1 in 48/5.
Chance that your 3rd number matches a winning lottery number is 1 in 47/4.
Chance that your 4th number matches a winning lottery number is 1 in 46/3.
Chance that your 5th number doesn't match a winning number is 1 in 45/(45-2) [because there are still 2 unmatched winning numbers].
Chance that your 6th number doesn't match a winning number is 1 in 44/(44-2) [yes, still 2 unmatched winning numbers].Now you need to accumulate all those chances by multiplying them together:
1 in (49/6)*(48/5)*(47/4)*(46/3)*(45/43)*(44/42) which is 1 in 15486.953.
Now this is the chance for that single case occurring, but there are 15 combinations of matching 4 from 6, so you divide the answer by 15 to get 1 in 15486.953/15 or 1 in 1032.4.
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3 Numbers (Constant prize: £10)
Follow exactly the same scheme as the 4 match above to get these figures:
1 in (49/6)*(48/5)*(47/4)*(46/43)*(45/42)*(44/41) (which is 1 in 1133.119) for a single case.
There are 20 combinations of 3 from 6, so the chance of a 3 match is 1 in 1133.119/20 or 1 in 56.7.
The chance of you winning any of the above prizes is approximately 54 to 1 - it is reckoned an average of one million people per draw will win a prize.
