Yes Mark, it works the same way here in Australia. I am just thinking about the best way of linking the 'prize' in the cake to the 'prize' in the lottery.

Wendy, the way the lottery works in the US is that if there is no winning number, the remaining money of the 'prize', called 'jackpot' is added to the receipts in the next draw. The cumulative 'jackpot' can become very high if there is no winner in successive draws. For the popular powerball lottery, the largest jackpot' ever was $2.04 billion, won on a single ticket in 1922. See https://www.nytimes.com/article/lottery-jackpot-record-powerball-megamillions.html

Mark, to better link the 'prize' in the cake with the 'prize' to be won in the lottery, how about using 'prize' instead of 'sum'.
If there is no winner, the prize is returned to play for the next draw.