The Hospital Research Foundation Home Lottery – calculation of chance of winning
Advice has been sought and received, based on this framework:
- A maximum of 230,000 tickets will be sold and will be eligible to be drawn;
- One ticket number is drawn at a time, after each draw the winning ticket number is recorded and is re-entered into the drawing process, which means that a ticket can win more than one prize;
- Each ticket number is equally likely to be drawn in a single draw, regardless of whether it has already won one or more prizes;
- there may be additional bonus draws that are available only to tickets purchased prior to certain dates;
- There is to be a 1 in 15 chance of winning a prize.
Calculations have been undertaken on the number of prize draws that would need to be made so that the chance of a ticket holder winning a prize is 1 in 15, on the assumption that all tickets are sold. If all tickets are not sold, the chance of winning will be more favourable than 1 in 15, for the calculated number of prize draws.
For the purposes of calculations, it has been assumed that the ticket that is purchased is not eligible for any of the bonus draws, which produces a more conservative calculation of the chance of winning (by removing the additional draws).
For a single draw, each ticket number has an equal chance of being drawn, meaning that there is a 1 in 230,000 chance of being drawn.
Initially, the calculation approach is on the basis of determining the chance of a ticket number NOT winning any prize. The complement of that outcome will then produce the chance of a single ticket winning a prize. This allows for the possibility that a single ticket can actually win more than one prize.
For any single draw, each ticket number has a 229,999 chance in 230,000 of not winning. For any individual ticket number to be “unlucky” in not winning a single prize, we need to multiply this probability together for the number of prize draws that are made. In mathematical terms, we are compounding the probability of “failure”.
If there is a 1 in 15 chance of winning a prize (excluding the additional prizes), the chance of failure must be 14 in 15 (or 0.933333), or less.
The actuary has calculated the probability of “failure” for a range of prize draws, and has calculated that 15,869 prize draws would be necessary to produce a probability of failure which is 0.933331. This means that the probability of “success” would be 0.066669, which is 1 in 14.99950, slightly more favourable than the required 1 in 15.
If there is one less prize draw (name 15,868), the probability of “failure” is 0.933335, and the complementary probability of “success” is 0.066665, which is less than 1 in 15.00040, which is slightly less favourable than the required 1 in 15.
This means that there needs to be at least 15,869 prize draws (excluding any bonus prize draws) to produce a 1 in 15 chance of winning a prize.
Please let us know if there any further questions you might have on the number of chances to win.
We thank you for your support and wish you all the best in all the draws. Good luck!