Sum is 3?

Probability Level 3

You have a fair coin with 1 on one side and 2 on the other side.

You keep flipping it, and keeping a running total. i.e. You start with a total of zero, and keep adding the number shown on the coin after every flip?

The probability that at some point the running total will be 3 3 is a b \dfrac{a}{b} , where a a and b b are coprime positive integers. What is a + b a+b ?


The answer is 13.

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Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Oct 23, 2016

Your first 3 3 flips can be:

  • 111
  • 112
  • 121
  • 122
  • 211
  • 212
  • 221
  • 222

Each is equiprobable, and further flips are irrelevant as the total will be more than 3 3 .

The second, seventh and eighth are the only ones that don't at some point lead to a total of 3 3 . The other 5 5 do.

Therefore the probability of having a total of 3 3 at some point is 5 8 \frac{5}{8}

5 + 8 = 13 5+8=\boxed{13}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Lots of food for though from seeing this problem.

How can we generalize this? What is the probability that (say) 20 is hit?
Does the probability tend to a particular value? If so, why?

Calvin Lin Staff - 4 years, 7 months ago

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Agreed! :) Lemme see if I can think of a good follow-up problem... Stay tuned!

Geoff Pilling - 4 years, 7 months ago

Here is my follow up question... :)

Geoff Pilling - 4 years, 7 months ago

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