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>
