Who Wants To Be A Riddler Millionaire?

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

Therefore the probability that $B$ is the correct answer after the lifeline is $0.875$.