Combinations and the EuroMillions lottery even when another Lucky Star is added. Firstly my level of mathematics understanding is only at O level or GCSE level if you like however my education level is just under degree level
having a HND in Computer Science.
Someone is saying that the overall chance of winning any prize on the EuroMillions is 1 in 13. Is this some mathematical "magic" number that applies to all lotteries? If so how can this be?
The Euromillions will shortly be adding another ball to the "Lucky stars"
so instead of 2 from 11 numbers ( 55 combinations ) they will be increasing it, as far as I Know, as 2 from 12 ( 66 combinations ).
There are currently 9 prize levels from the 13 available that involve
a ticket winning any money back where 1 or 2 "Lucky star" nymbers are involved.
The main ball numbers will remain as any 5 from 50.
So by my understanding, any lottery system that increases the odds of winning for most of the prize tier levels can not surely maintain an average win rate of 1 in 13 tickets, can they? Not unless some how they "FIX" the system so that some winning tickets and some losing tickets are secretly printed after the draw has been made.
Your thoughts please or is this " 1 in 13 " to do with some higher level mathematical understanding on betting systems and probability?
 A: All prizes in EuroMillions are awarded for tickets matching at least:


*

*one main number and both Lucky Stars (A) or

*two main numbers (B)


To find the probability of winning any prize, we can calculate $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, the formula following from the inclusion/exclusion principle. Since the state space from which the lottery numbers are drawn is constant, the event probabilities can be replaced by the number of ways they can occur.
In the new format, the main numbers can be drawn in $\binom{50}5=2118760$ ways; the Lucky Stars can be drawn in $\binom{12}2=66$ ways, as you mentioned. Multiplying, we get 139838160 possible draws.
There is obviously only one way to match both Lucky Stars. Now in order to not match any of the draw's numbers, the ticket must have 5 numbers from the 45 not drawn, and there are $\binom{45}5=1221759$ ways this can happen. Hence the ticket has $\binom{50}5-\binom{45}5=897001$ ways of matching at least one main number, and event A has 897001 matching tickets.
To match at least two main numbers, note that for the ticket to match exactly one number it must choose one of the draw's five numbers to match, then pick its remaining four numbers from the 45 not drawn. Hence there are $897001-5\binom{45}4=152026$ ways the ticket can match at least two main numbers. For event B there are no restrictions on the Lucky Stars, so by multiplication there are 10033716 ways it can occur.
$A\cap B$, representing the tickets matching both Lucky Stars (1 way) and at least two main numbers (152026 ways), has simply 152026 ways of happening. Therefore, we conclude that for a given draw there are
$$897001+10033716-152026=10778691$$
tickets winning any prize. The probability a ticket wins any prize is thus
$$\frac{10778691}{139838160}=\frac{46661}{605360}\simeq\frac1{12.97}$$
Indeed, this is close to 1-in-13, as claimed. Even more so, the chances of winning any prize in the old format are approximately 1-in-12.80 – almost exactly the same.

By adding an extra ball the odds of winning have indeed decreased, but not by much; while the state space has been enlarged, the number of winning tickets has increased almost in tandem. It is the jackpot that has truly become harder to strike, and accordingly the minimum jackpot has been raised to £14 million (inflation is another cause for this pay rise).
The 1-in-13 chance of winning any prize on EuroMillions is of course not constant across lotteries around the world. For example, the American Mega Millions has around a 1-in-14.7 chance of winning any prize.
