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Camilla's electronic lottery
 

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The first chart shows the probability of having a given relative balance, compared with the other possible balances.

The second chart shows the probability of having at least a given relative balance – or a greater amount.

A relative balance is the total of prize amounts minus the cost of ticket purchases required to play (in this case, for the selected number of years/draws).  It is a comparison with having not played at all, where non-participation has a zero relative balance.  If ticket costs are not covered by prizes, then the relative balance will be negative.

Odds of winning the jackpot prize (£100)

The probability of matching all 5 numbers is 1 in 252 (around 0.4%), which corresponds to winning the jackpot prize approximately once every 5 years on average.  The calculation is as follows.  For the first winning number, there are 10 possibilities and you have 5 choices on your ticket – so you have a 5⁄10 chance of a match.  For the second selection, there are 9 remaining numbers for the draw and 4 unmatched numbers on your ticket, so have a 4⁄9 chance of a match – and so on for the other numbers.  Multiplying all of the probabilities together gives the jackpot prize odds as follows.
5⁄10 x 4⁄9 x 3⁄8 x 2⁄7 x 1⁄6 = 120⁄30,240 = 1⁄252

Odds of winning the popular prize (£10)

The probability of matching 4 numbers is nearly 1 in 10 (around 9.9%).  The calculation for this is slightly more complicated than that for the jackpot prize, shown above, and is worked out by taking the probability of matching those numbers multiplied by the probability of not matching the remaining number.  That result then has to be multiplied by the various permutations – as you could match the first 4 numbers, the last 4, or some other combination in between – there are 5 variations in all.
( 5⁄10 x 4⁄9 x 3⁄8 x 2⁄7 x 5⁄6 ) × 5 = ( 600⁄30,240 ) × 5 = 25⁄252 = 1⁄10.08

Odds of winning the small prize (£1)

The probability of matching exactly one number is also nearly 1 in 10 (around 9.9%).  It is calculated in the same way as for matching 4 numbers, shown above, and gives the same result.  The only difference is in the order of the figures, where the first probability is that of matching the first number and the remaining probabilities are for not matching the other 4 numbers.  Again, there are 5 different ways in which matches might take place.
( 5⁄10 x 5⁄9 x 4⁄8 x 3⁄7 x 2⁄6 ) × 5 = ( 600⁄30,240 ) × 5 = 25⁄252 = 1⁄10.08

Odds of matching 2 or 3 numbers (no prizes)

The probability of matching exactly two or three numbers is 25⁄63 (around 39.7% for each, or 79.4% combined).  The calculations are like those for matching 1 or 4 numbers, shown above, except there are now 10 different combinations of positions in which the 2 or 3 matching numbers might be found.
Matching 2 numbers
( 5⁄10 x 4⁄9 x 5⁄8 x 4⁄7 x 3⁄6 ) × 10 = ( 1,200⁄30,240 ) × 10 = 25⁄63 = 1⁄2.52

Matching 3 numbers
( 5⁄10 x 4⁄9 x 3⁄8 x 5⁄7 x 4⁄6 ) × 10 = ( 1,200⁄30,240 ) × 10 = 25⁄63 = 1⁄2.52

Odds of winning the mystery prize

The probability of winning the mystery prize is a secret.  It is intended as an extra game, for the players to work out the circumstances in which the mystery prize can be won – hopefully before they actually win it!