Please help with this probability Q

Question

1. Ally is a contestant on a TV game

show called Win a Million. Each time

she answers a multiple-choice question

correctly, she wins money. If she picks a

wrong answer, she is eliminated. If Ally does

not know the right answer, she can use one

of the following Helping Hands:

• Quiz the Crowd: She can poll the audience.

The crowd has an experimental probability

of being correct 85% of the time.

• Double Up: She can give two answers,

instead of just one. If either is correct she

stays in the game.

• Rule One Out: One of the incorrect

answers is removed, leaving three choices.

Suppose Ally encounters three questions in a

row to which she does not know the answers.

a) Assuming that she can use each Helping

Hand only once during the game, and

only once per question, what is the best

estimated probability Ally has of staying alive through the three questions? What

assumptions did you make.

b) How many more times is Ally likely to

stay in the game if she uses all three

Helping Hands than if she simply guesses

at random on all three questions?

Answer 1

For DoubleUp the probability of correct answer is 1/2 for the following reason.

By choosing 2 answers she partitions the given answers into two equal partition.

The correct answer has equal chance of being in either partition.

For RuleOneOut the probability of correct answer is 1/3, because

one of the three remaining choices is correct.

Not using any helping hand gives her probability 1/4, because

she needs to chose one of four possibilities.

a) Her best probability is to use a Helping Hand for each question,

because the alternative of a random guess has worse odds.

Her probability of staying in the game is then 0.85 × 1/2 × 1/3 = 0.142.

I am assuming that

- Any of her guesses has equal chance of being right.

- The audience has the same chance of 85% for all three questions.

- Knowing the correct answer to one question does not provide any hint for others.

b) If she simply guesses all three questions then her chances of staying in the game are

1/4 × 1/4 × 1/4 = 0.015625,

which is about 9 times worse than using the Helping Hands.