Who wins the Dice Game?
Level
2
Phineas and Ferb play a game of rolling a dice. Phineas starts first and they take alternate turns. Whosoever gets the first 6 on the dice wins. What is the probability that Phineas Wins this game?
11 / 20
6 / 10
6 / 11
1 / 2
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1 solution
The answer is 6 / 11 , contrary to the 1 / 2 that we might expect at first glance. The important thing is that Phineas starts first . So how does that make a difference?
The probability of Phineas winning in the first go is 1 / 6
If he doesn't win, then Ferb must also not win so that Phineas can win in his second chance.
The probability of Phineas winning in the second go is (5 / 6) x (5 / 6) x (1 / 6)
where (5 / 6) means that a six was not obtained.
The same way, the probability of Phineas winning in the third go is (5 / 6) x (5 / 6) x (5 / 6) x (5 / 6) x (1 / 6)
We recognize that the sequence of probabilities of Phineas winning becomes:
(1 / 6), (25 / 216), (625 / 7776)...
we then recognize that this is a geometric progression with a common ratio (R): (25 / 36)
Applying the formula for the sum of infinite geometric progression:
(1 / 6) x (1 / (1 - R)) gives me 6 / 11 !!