What Are The Chances of Getting a Six (and what does that have to do with the lottery anyway!)?

[Q] My friend reckons if he buys one ticket, his odds of winning the lottery jackpot are about 1-in-14 million. He says if he buys just one ticket in his lifetime, the odds are exactly the same as if he wastes money on buying a ticket every week. Is he right?

[A] No.

Simplify it to dice. Always seems easier with dice :-)

Let's say somebody was offering you and your friend a million bucks if you could throw a 6. Only snag is, one of you gets to have 1,000 throws of the dice, the other only gets one throw. Would your friend really let you have the 1,000 throws without the slightest complaint..?

The chances of getting a 6 in one throw are 1/6.

For two throws, there are three ways to get a six:
[1] First throw IS a six, second throw NOT a six
[2] First throw NOT a six, second throw IS a six
[3] Both throws are a six.

Chances of [1] are 1/6 x 5/6 = 5/36
(i.e. 1-in-6 chance of getting the 6, then 5-in-6 of getting anything but a six)
Chances of [2] are 5/6 x 1/6 = 5/36
Chances of [3] are 1/6 x 1/6 = 1/36

Total chance of getting at least one six in two throws of the dice is therefore 11/36, or just over 30%.

The formula for rolling at least one 6 in n rolls is: 1 - (5/6)^n

So even just 20 throws gives you a 97% chance of a 6 :-)