Example Question - keys
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Probability of Successfully Opening a Lock with Multiple Keys
<p>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:</p> <p>\[1 - \frac{1}{3} = \frac{2}{3}\]</p> <p>(a) The probability that Asri fails to open the lock on her first attempt:</p> <p>\[\frac{2}{3}\]</p> <p>(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:</p> <p>\[n = 3 + 2 = 5\]</p> <p>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:</p> <p>\[\frac{1}{5} \times \frac{1}{3}\]</p> <p>And the probability she fails to open the lock with the new keychain is:</p> <p>\[1 - \frac{1}{5} \times \frac{1}{3}\]</p> <p>\[\Rightarrow 1 - \frac{1}{15}\]</p> <p>\[\Rightarrow \frac{15}{15} - \frac{1}{15}\]</p> <p>\[\Rightarrow \frac{14}{15}\]</p> <p>The new probability that the lock is successfully opened is:</p> <p>\[\frac{1}{15}\]</p>
Probability of Failing and Succeeding in Opening a Lock with Multiple Keys
<p>(a) The probability that Asri fails to open the lock:</p> <p>P(\text{failure}) = 1 - P(\text{success})</p> <p>P(\text{failure}) = 1 - \frac{1}{3}</p> <p>P(\text{failure}) = \frac{2}{3}</p> <p>(b) The probability that Asri fails to open the lock with the new key added:</p> <p>Since the two new keys cannot open the lock, the total number of keys is now 5.</p> <p>P(\text{new failure}) = \frac{\text{Number of keys that do not work}}{\text{Total number of keys}}</p> <p>P(\text{new failure}) = \frac{4}{5}</p> <p>(c) The probability that the lock is successfully opened with the new key added:</p> <p>P(\text{new success}) = 1 - P(\text{new failure})</p> <p>P(\text{new success}) = 1 - \frac{4}{5}</p> <p>P(\text{new success}) = \frac{1}{5}</p>
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