Question

here is the setup for a non - traditional casino game: you draw a card from a well shuffled full deck and if the card is a king you win $100. the game costs $2 to play and you decide to play the game until you win the $100. each time you draw a card you pay $2, and if the card is not a king, the card is put back in the deck, and the deck is reshuffled. how much money should you expect to spend on this game? 26 100 4 52 13

Explanation:

Step1: Calculate the probability of drawing a king

A standard deck has 52 cards and 4 kings. The probability \(p\) of drawing a king is \(p=\frac{4}{52}=\frac{1}{13}\).

Step2: Use the formula for the expected number of trials in a geometric distribution

The expected number of trials \(E(X)\) in a geometric distribution (where we repeat an experiment until we get a success) is \(E(X)=\frac{1}{p}\). Substituting \(p = \frac{1}{13}\), we get \(E(X)=13\).

Step3: Calculate the expected cost

Each trial (drawing a card) costs \( \$2\). If the expected number of trials is \(n = 13\), then the expected cost \(C\) is \(C=2\times E(X)\). Substituting \(E(X) = 13\), we have \(C = 2\times13=\$26\).

Answer:

26