[Math] Players $A$ and $B$ alternately toss a biased coin,with $A$ going first.$A$ wins if $A$ tosses a tail before $B$ tosses a head;otherwise $B$ wins.

Question

Players $A$ and $B$ alternately toss a biased coin,with $A$ going first.$A$ wins if $A$ tosses a tail before $B$ tosses a head;otherwise $B$ wins.If the probability of a head is $p$,find the value of $p$ for which the game is fair to both players.

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I do not understand what its mean by the game is fair to both players.Does it mean the probability of winning is same for both the players$?$

Answer 1

$A$ wins on the first toss if $T$, probability $(1-p)$

$B$ wins on the second toss if $HH$, with probability $p^2$

If neither wins in two tosses, we are back to the start.

Thus for both to have equal chances, $(1-p) = p^2$

Proceed....

$0.5(\sqrt5 - 1)$