Question - Probability of Failing and Succeeding in Opening a Lock with Multiple Keys

(a) The probability that Asri fails to open the lock:

P(\text{failure}) = 1 - P(\text{success})

P(\text{failure}) = 1 - \frac{1}{3}

P(\text{failure}) = \frac{2}{3}

(b) The probability that Asri fails to open the lock with the new key added:

Since the two new keys cannot open the lock, the total number of keys is now 5.

P(\text{new failure}) = \frac{\text{Number of keys that do not work}}{\text{Total number of keys}}

P(\text{new failure}) = \frac{4}{5}

(c) The probability that the lock is successfully opened with the new key added:

P(\text{new success}) = 1 - P(\text{new failure})

P(\text{new success}) = 1 - \frac{4}{5}

P(\text{new success}) = \frac{1}{5}