What is the correct key?

Probability Level 2

A person with a bunch of nine keys is to open a door but only one key can open. What is the probability that he will succeed in three trials?

Note: Assume that it is dark that he cannot see the correct key.

1 2 \dfrac{1}{2} 1 9 \dfrac{1}{9} 1 84 \dfrac{1}{84} 1 3 \dfrac{1}{3}

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2 solutions

Siva Budaraju
Oct 3, 2017

The answer is just 1 9 \dfrac{1}{9} + + 1 9 \dfrac{1}{9} + + 1 9 \dfrac{1}{9} = 3 9 \dfrac{3}{9} = 1 3 \boxed{\dfrac{1}{3}} because each of the keys has a 1/9 probability of opening the door, and you are trying out 3 keys. By the Addition Rule of Probability, the answer is the sum of the three probabilities, or 1 3 \boxed{\dfrac{1}{3}} .

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The possibilities are o p e n , n o t o p e n − o p e n , n o t o p e n − n o t o p e n − o p e n open, not~open-open, not~open-not~open-open .

P = 1 9 + 8 9 ( 1 8 ) + 8 9 ( 7 8 ) ( 1 7 ) = P=\dfrac{1}{9}+\dfrac{8}{9}\left(\dfrac{1}{8}\right)+\dfrac{8}{9}\left(\dfrac{7}{8}\right) \left(\dfrac{1}{7}\right)= 1 3 \boxed{\dfrac{1}{3}}

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