Hi,

Almost 10 years ago I wrote a post comparing the probability of winning the National Lottery to the number of hours in a lifetime. I've now re-done the calculations to get up-to-date with the changes to the National Lottery and British life expectancy.

The purpose is to show how incredibly unlikely it is to win the lottery by using a comparison between the probability of winning and the number of hours in a lifetime. Quite effective I think at spoiling the idea of 'It Could be You' (their advertising slogan) or at least putting how unlikely it is to be you into perspective.

Could forum members check my methodology please - many thanks. [Note: the arithmetic has been triple checked.]

Probability of winning the:Lottojackpot

Theflagshipgame in the British National Lottery is theLottogame which has two draws per week, on Wednesdays and Saturdays. Players choose 6 numbers in the range from 1 to 59 (increased from 49 in 2015) and matching all 6 of them wins theLottojackpot (or an equal share if more than one winning ticket).

$\dfrac{59!}{6!\cdot53!}=45057474$

So there's a 1 in 45,057,474 chance of winning (as stated on Wikipedia).

How many plays of the lottery would it take to have a 99% chance of winning at least once?

$P(\text{winning at least once in n attempts}) = 1 - P(\text{not winning in 1 attempt})^n$

$P(\text{not winning in 1 attempt}) = 1 - \dfrac{1}{45057474} = \dfrac{45057473}{45057474}$

Let n represent the number of times it would be necessary to play the lottery to have a 99% chance of winning at least once.

$1 - \left(\dfrac{45057473}{45057474}\right)^n = \dfrac{99}{100} \longrightarrow \left(\dfrac{45057473}{45057474}\right)^n = \dfrac{1}{100} \longrightarrow n \cdot \log \left(\dfrac{45057473}{45057474}\right) = \log \left(\dfrac{1}{100}\right) \longrightarrow n \approx 207497333.8261$

Approx. 207,497,334 plays are needed to have a 99% probability of winning at least once.

Equating the probability of winning theLottojackpot to hours in a lifetime, etc..

British Life Expectancy: Male: 79.2; Female: 82.9; Mean Average: 81.05 (Source: Office of National Statistics 2017)

1) Playing once every hour (if there were 24 draws per day instead of 2 per week):

$\dfrac{207497333.8261}{365.25 \cdot 24} \approx 23670.6974 \hspace{3em} \dfrac{23670.6974}{81.05} \approx 292.0506$

Playing once every hour, i.e. 24 times a day, you would have to live to be about 23,670 years old to have a 99% probability of winning at least once. That's about 292 lifetimes of 81 years.

2) Playing once a week:

$\dfrac{207497333.8261}{365.25 \div 7} \approx 3976677.1712 \hspace{3em} \dfrac{3976677.1712}{81.05} \approx 49064.4932$

Playing once a week you would have to live to be about 3,976,677 years old to have a 99% probability of winning at least once. That's about 49,064 lifetimes of 81 years.

3) Playing twice a week (both weekly draws):

$\dfrac{207497333.8261}{365.25 \div 3.5} \approx 1988338.5856 \hspace{3em} \dfrac{1988338.5856}{81.05} \approx 24532.2467$

Playing twice a week (the draw on Wednesdays and Saturdays) you would have to live to be about 1,988,338 years old to have a 99% probability of winning at least once. That's about 24,532 lifetimes of 81 years.