1. You are stuck on a desert island with a friend and single coin which is your

Question

1. You are stuck on a desert island with a friend and single coin which is your only source of randomness. [Deterministic desert islands are very boring!] The coin in question is biased in that it comes up heads more often than tails, though you do not know the exact bias. (It does come up tails some of the time.) You and your friend want to play a game that needs a fair coin. How can you use the biased coin to simulate the fair coin that you need? [5] 2. Having been rescued, you and your friend get stuck on another desert island with a single coin, but this time it’s a coin that is known to be fair. Having gotten bored of the game needing a fair coin on the last desert island adventure, you and your friend would like to play a game that requires a biased coin, in particular, a coin with P(H) = 0.6 and P(T) = 0.4. How can you use the fair coin to simulate the biased coin you need? [5]

Explanation / Answer

1) Use biased as fair-

(H, T): The probability to get 0 followed by 1 from two throws = 0.6 * 0.4 = 0.24
(T, H): The probability to get 1 followed by 0 from two throws = 0.4 * 0.6 = 0.24

So the two cases appear with equal probability. The idea is to return consider only the above two cases, return 0 in one case, return 1 in other case. For other cases [(T, T) and (H, H)], recur until you end up in any of the above two cases.

b) Throw a coin thrice .

THere are eight possible outcomes.

HHH

HTH

HHT

HTT

TTH

THT

TTT

Use any 3 of these 3 outcomes as ;Head' and any other two outcomes as tails. The rest 3 outcomes can be discarded.

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