Expert: Ahmed Salami - 5/28/2010

Question
I was asked recently to calculate the number of times I would have to roll a fair die to guarantee a 50% chance of rolling a 1
I know, from my own studies, that The general formula for rolling at least one 1 in n rolls is 1 - (5/6)^n
Computationally, I determined that the answer is between 3(42%) and 4(52%) but is there a way to solve for n?
(For example, a lottery has 15,346 tickets, how many times would I have to buy one to give me a 50% chance overall of winning, or a 99% chance, or a 1/15,346 chance. There must be a general formula.... or not...?

Answer
Hi Patrick,
Suppose you wanted to solve the equation;
1 - (5/6)^n = 50% = 0.5    
then
(5/6)^n = 0.5
we take logarithms of both sides
log(5/6)^n = log(0.5)
n.log(5/6) = log(0.5)
n = log(0.5)/log(5/6)
 = log(0.5)/log(0.833)
 = -0.3010/-0.0792
 = 3.8

Of course you cant roll a die 3.8 times, but you get the idea and stick with 4.

Regards