Two players A and B, toss a fair coin. A, who starts the game. stakes a penny each time he throws the coin; B also stakes a penny at each of his throws. The first player to throw heads wins and gathers all the stakes. Find the probability that A wins the game. Explain why A, although he wins more frequently, loses money. Calculate his expected loss on 100 games
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P[A wins (can only be on odd throws)]
= 1/2 + 1/8 + 1/32 + ..... , a GP with a = 1/2, r = 1/4,
Soo = a/(1-r) = (1/2)/(3/4) = 2/3
revised as per your AD
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positive net winnings (W) for A start only from 3rd throw (on 1st throw only the stake gets returned).
on the 3rd throw, has put in 2, gets 3, net winnings 1
on the 5th throw, has put in 3, gets 5, net winnings 2 & so on
W = 1*1/8 + 2*1/32 + 3*1/128 + ........
W/4 = ....... 1*1/32 + 2*1/128 + .....
subtracting, 3W/4 = 1/8 + 1/32 + 1/128 + ..... = (1/8)/(3/4) = 1/6
thus W = 1/6 *4/3 = 2/9
but there will be net losses (L) on the games lost
L = 1*1/4 + 2*1/16 + .... = 4/9
expected *loss* in 100 games = 100(4/9 - 2/9), ≈ 22
all is well that ends well !
= 1/2 + 1/8 + 1/32 + ..... , a GP with a = 1/2, r = 1/4,
Soo = a/(1-r) = (1/2)/(3/4) = 2/3
revised as per your AD
-----------------------------
positive net winnings (W) for A start only from 3rd throw (on 1st throw only the stake gets returned).
on the 3rd throw, has put in 2, gets 3, net winnings 1
on the 5th throw, has put in 3, gets 5, net winnings 2 & so on
W = 1*1/8 + 2*1/32 + 3*1/128 + ........
W/4 = ....... 1*1/32 + 2*1/128 + .....
subtracting, 3W/4 = 1/8 + 1/32 + 1/128 + ..... = (1/8)/(3/4) = 1/6
thus W = 1/6 *4/3 = 2/9
but there will be net losses (L) on the games lost
L = 1*1/4 + 2*1/16 + .... = 4/9
expected *loss* in 100 games = 100(4/9 - 2/9), ≈ 22
all is well that ends well !