Basic probability! using the lottery as means of explaining odds of winning prize

[ukdad]

Hi guys, new to the forum.

Just wondered if anyone could help me with some basic probability. I used to think I was pretty good at maths, but I was recently helping my daughter with her maths homework and found I was confusing myself and questioning what I was trying to teach her.. I guess modern technology has made my brain lazy over the past 2 decades!

In basic terms, I was using the lottery as a means of explaining the odds of winning a prize. Using the UK Thunderball game as an example shown here https://www.national-lottery.co.uk/games/thunderball/game-procedures I was explaining the 1 in 111 chances of matching 3 numbers to win £10. My daughter then asked.. if you bought 111 tickets would you be guaranteed a £10 prize? To which I confidently answered 'yes, assuming each ticket held a different combination of numbers'

I have since questioned my response, as it doesn't logically add up in my head, and I've struggled to find a simple equation to prove or disprove it.

In the game mentioned above, you chose a line of 5 numbers between 1 - 39. There must be a simple way of working out how many lines would guarantee matching 3 numbers?

I know this will be very basic compared to the standard you guys are at here, but can someone kindly help me put my mind at rest here!? Feeling a bit dumb!

 

[pka]

I have since questioned my response, as it doesn't logically add up in my head, and I've struggled to find a simple equation to prove or disprove it. In the game mentioned above, you chose a line of 5 numbers between 1 - 39. There must be a simple way of working out how many lines would guarantee matching 3 numbers?

I confess that I do not follow what you really want.
But there are 575757 ways to select five numbers \(\displaystyle 1\to 39\)
There is a probability of \(\displaystyle \dfrac{1}{575757}\) of exact match.
Given any selection of five numbers there are ten ways of selecting three. There are 11220 ways for the lottery’s five-selection to match exactly three of your numbers.
That is a probability of \(\displaystyle \dfrac{11220}{575757}\).

 

[ukdad]

Thanks for your reply. My apologies, I didn't explain it too well.

According to the website there is a 1:111 chance of matching 3 of 5 numbers (paying £10).

My daughter asked if that meant that buying 111 different tickets would guarantee a match of 3 from 5 numbers.. I initially said yes, but having given it some thought, that wouldn't seem correct.

In a nutshell, how many tickets are required to guarantee a match of 3 from 5 numbers (of numbers between 1- 39)? And how does one get to the answer?