consider the following game orignally from nicholas bernoulli: each of two playe

Question

consider the following game orignally from nicholas bernoulli: each of two players, alice and bob, has a deck with the same cards. they draw cards singly until each draws the same card, in which case alice wins. if such a match does not occur and they exhaust the decks, then bob wins.

a) consider the game with 3 cards, and say they are labeled "1", "2", "3". what is the propability that bob wins the game? (hint: you can safely assume alice drew her cards in their natural order, that is alice drew cards as 1, 2, 3.)

b) consider the game with 4 cards, and say they are labeled "1", "2", "3", "4". what is the probability that bob wins this game?

Explanation / Answer

a) Considering the game with 3 cards labelled "1", "2", "3".

Assuming that alice drew her cards in their natural order, that is 1, 2, 3. Suppose alice draws 1, then bob can draw either 1 or 2 or 3. So for every card that alice draws, the chances of a card to be drawn by bob are 3. So, the total number of cases are 9 as can be seen in the below table.

For alice to win the game, the number on their cards should match. So, out of the 9 cases, the number of chances for alice to win the game are 3.

So, the probability that alice wins the game =

(Number of chances for alice to win the game) / (Total number of cases) = 3/9 = 1/3

Probability that bob wins the game = 1 - (Probability that alice wins the game) = 1 - 1/3 = 2/3

b) Considering the game with 3 cards labelled "1", "2", "3", "4".

Assuming that alice drew her cards in their natural order, that is 1, 2, 3, 4. Suppose alice draws 1, then bob can draw either 1 or 2 or 3 or 4. So for every card that alice draws, the chances of a card to be drawn by bob are 4. So, the total number of cases are 16 as can be seen in the below table.

For alice to win the game, the number on their cards should match. So, out of the 16 cases, the number of chances for alice to win the game are 4.

So, the probability that alice wins the game =

(Number of chances for alice to win the game) / (Total number of cases) = 4/16 = 1/4

Probability that bob wins the game = 1 - (Probability that alice wins the game) = 1 - 1/4 = 3/4

Card drawn by alice Card drawn by bob 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3

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