Heads and two tails?

Probability Level 3

Ralph Malph flips a fair coin until he has flipped heads at least once and tails at least twice.

What is the expected number of times he will need to flip the coin?

Image credit: http://mt4trader.net/coin-toss-ea.html


The answer is 4.5.

Geoff Pilling by Geoff Pilling , 53, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Geoff Pilling
Jul 15, 2017

Let's define the following:

E = expected value before any flips T = The expected value once you have only flipped at least one tail and no heads. H = The expected value once you have only flipped one head and no tails. HT = The expected value once you have only flipped at least one head tails and only one tails. TT = The expected value once you have at least two tails but no heads.

So, this can be solved with the following set of linear equations:

  • E = 1 + 0.5 ∗ ( H + T ) E = 1 + 0.5*(H+T)
  • T = 1 + 0.5 ∗ ( H T + T T ) T = 1 + 0.5*(HT+TT)
  • H = 1 + 0.5 ∗ ( H + H T ) H = 1 + 0.5*(H + HT)
  • H T = 1 + 0.5 ∗ H T HT = 1 + 0.5*HT
  • T T = 1 + 0.5 ∗ T T TT = 1 + 0.5*TT

Solving:

E = 4.5 E = \boxed{4.5}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...