Question - Probability of Successfully Opening a Lock with Multiple Keys

The first part of the problem states that there is a \(\frac{1}{3}\) probability of opening the lock when Asri chooses a key randomly. Therefore, the probability that Asri fails to open the lock is:

\[1 - \frac{1}{3} = \frac{2}{3}\]

(a) The probability that Asri fails to open the lock on her first attempt:

\[\frac{2}{3}\]

(b) Asri has two new keys added to her keychain that cannot open the lock. Let \(n\) be the total number of keys after adding two ineffective keys, then:

\[n = 3 + 2 = 5\]

The probability that she chooses a key that can open the lock is still \(\frac{1}{3}\), since the number of working keys has not changed. The total number of keys is 5 now, so the probability she chooses a working key is:

\[\frac{1}{5} \times \frac{1}{3}\]

And the probability she fails to open the lock with the new keychain is:

\[1 - \frac{1}{5} \times \frac{1}{3}\]

\[\Rightarrow 1 - \frac{1}{15}\]

\[\Rightarrow \frac{15}{15} - \frac{1}{15}\]

\[\Rightarrow \frac{14}{15}\]

The new probability that the lock is successfully opened is:

\[\frac{1}{15}\]