Riddler Express
You’ve made it to the \$1 million question, but it’s a tough one. Out of the four choices, \(A\), \(B\), \(C\) and \(D\), you’re \(70\) percent sure the answer is \(B\), and none of the remaining choices looks more plausible than another. You decide to use your final lifeline, the \(50:50\), which leaves you with two possible answers, one of them correct. Lo and behold, \(B\) remains an option! How confident are you now that \(B\) is the correct answer?
Solution
Let \(B\) be the event that \(B\) is the correct answer. Let \(L\) be the event that the two options (one of which is the correct answer) from lifeline contains \(B\).
We have \(\mathbb{P}[B] = .7\) and we are interested in \(\mathbb{P}[B \| L ]\).
From Bayes Theorem, we have
\[ \begin{equation*} \mathbb{P}[B \| L] = \frac{\mathbb{P}[L \| B] \mathbb{P}[B]} {\mathbb{P}[L \| B] \mathbb{P}[B] + \mathbb{P}[L \| B'] \mathbb{P}[B']} = \frac{1 \cdot 0.7}{1 \cdot 0.7 + \frac{1}{3} \cdot 0.3} = \frac{7}{8} = 0.875 \end{equation*} \]
Therefore the probability that \(B\) is the correct answer after the lifeline is \(0.875\).