An unfair coin

Probability Level 4

For an unfair coin, if E ( H H ) = E ( T ) E(HH) = E(T) , then what is E ( T ) E(T) to 3 decimal places?

Clarifications :

  • E ( H H ) E(HH) denotes the expected value for the number of flips you would make before flipping 2 heads in a row

  • E ( T ) E(T) denotes the expected value for the number of flips you would make before flipping a tail.

Other Expected Value Quizzes
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The answer is 3.414.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Jul 6, 2016

Relevant wiki: Expected Value - Problem Solving

E ( T ) = 1 / P T E(T) = 1/P_T where P T P_T = probability of flipping tails

E ( H H ) = ( 2 − P T ) / ( 1 − 2 P T + P T 2 ) E(HH) = (2-P_T)/(1-2P_T+P_T^2)

This can be derived by solving the following set of linear equations, where E i E_i represents the expectation value of getting i i n n 's once you have i i n n 's :

  • E = 1 + P T × E + ( 1 − P T ) E 1 E = 1 + P_T \times E +(1-P_T)E_1
  • E 1 = 1 + P T × E E_1 = 1 + P_T \times E

Setting them equal, gives that P T = 0.293 P_T = 0.293 .

So E ( T ) = E ( H H ) = 3.414 E(T) = E(HH) = \boxed{3.414}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Hi how did you get the equation for E(HH)? Thank you!

Ashish Sacheti - 4 years, 11 months ago

This can be derived by solving the following set of linear equations, where E i E_i represents the expectation value of getting i i n n 's once you have i i n n 's :

  • E = 1 + P T × E + ( 1 − P T ) E 1 E = 1 + P_T \times E +(1-P_T)E_1
  • E 1 = 1 + P T × E E_1 = 1 + P_T \times E

Make sense?

Geoff Pilling - 4 years, 11 months ago

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