Who Wants To Be A Riddler Millionaire?

A FiveThirtyEight Riddler puzzle.

By Vamshi Jandhyala in mathematics Riddler

October 4, 2019

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 $0.875$.