# Who Wants To Be A Riddler Millionaire?

A FiveThirtyEight Riddler puzzle.
mathematics
Riddler
Published

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 $$0.875$$.