A carnival game involves drawing a card from a deck of 40 cards, replacing it, shuffling the deck, and drawing another card. Thirty of the cards are blank and ten are labeled 8 tokens. Your prize is the sum of the cards you draw. What is the
expected payoff for this game?
4 tokens
2 tokens
8 tokens
O tokens

4 tokens.

Step-by-step explanation:

The expected value of a given event is calculated as:

EV = (x₁*p₁ + x₂*p2 + ... + xₙ*pₙ)

Where xₙ is the n-th outcome, and pₙ is its probability.

In this case, our experiment is:

You draw two times.

We have 30 cards with no prize

We have 10 cards with a prize.

A total of 40 cards.

As we draw two times (and the first time we draw a card we put it back in the deck) we can consider the events as independent, so we can find the expected value per draw.

Now we can define:

x₁ = drawing a blank card = 0 tokens

The probability will be equal to the quotient between the number of blank cards and the total number of cards

p₁ = 30/40 = 3/4

x₂ = drawing a prized card = 8 tokens.

The probability will be equal to the quotient between the number of prized cards and the total number of cards:

p₂ = 10/40 = 1/4

Then the expected value per draw is:

EV = ( (3/4)*0 tokens + (1/4)* 8 tokens) = 2 tokens.

And we have two draws, then the expected value of two draws is two times the expected value per draw, this means that the expected value in our case is:

expected value = 2*(2 tokens) = 4 tokens.

The correct option is the first one, counting from the top.

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