Name ______
Probability and Expected Value Task
You have won a chance to be on a game show! The goal will be to walk away from the game with as much money as possible.
1. The contestant begins with $100 to play the game.
2. The host shuffles a standard deck of playing cards with two jokers included.
3. Each round, the contestant pulls a card without looking at the cards in the deck and reveals the card.
4. If the card is black, wins $1 times the value of the card. An ace has a value of 1 and any face card (Jack, Queen, and King) has a value of 10. All other numbers have the value of the number on the card. The host then hands over the money to the contestant.
However, if the card is red, the contestant loses $1 times the value of the card. The contestant returns the money lost to the host.
SPECIAL BONUS: If the contestant pulls out a Joker, he immediately rolls a 6-sided die and wins $2 times the number rolled on the die. The host hands the amount won over to the contestant.
5. The contestant replaces the card revealed, and the dealer mixes the cards again. The contestant then chooses whether to quit and take the money he was remaining or to pull another card.
6. Repeat steps 4 and 5 until the contestant chooses to quit or pulls a total of 10 cards. The contestant walks away with whatever money he has remaining.
Let’s choose a winner to come up front and play the game!
Now, split up into pairs. Take turns playing the game with one person being the contestant and the other the host. When you are the contestant record your results on the table below.
Round / Card / Amount Won / Total Amount Remaining1
2
3
4
5
6
7
8
9
10
We will take a few minutes to compare and discuss our results. If you finish early, feel free to repeat the game for fun to try to find out how much money you can win. You only need to record the results from the first time you play.
Let’s take a look at the statistics behind the game.
1. Complete the following tables by writing the dollar amount that the contestant would win for pulling each card. Remember that red cards end up with a negative result.
2. Write the probability of each possible result in the next row labeled probability. Remember that the deck has a total of 54 cards total. Also, in order to find the probability for the die rolls you may have to consider compound probabilities.
3. Complete the final row of the tables by finding the products of the two numbers above each empty box. For example, to fill in the first product box multiply the probability of revealing a black ace (1/27) by the amount the contestant wins for the card ($1). Then, write the answer (1/27)($1) = $0.04. Why did I write the answer as a decimal rounded to two decimal places?
4. Add all of the products together in the final row of each table to find the sum of the products.
5. Calculate the expected value by adding all of the sums of products together. Then, answer the questions that follow.
Black / Ace / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / Jack / Queen / KingAmount Won
Probability
Product
Sum of the Black Card Products = ______
Red / Ace / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / Jack / Queen / KingAmount Won
Probability
Product
Sum of the Red Card Products = ______
Joker and Die Roll / 1 / 2 / 3 / 4 / 5 / 6Amount Won
Probability
Product
Sum of the Die Roll Products = ______
Now, calculate the expected value, E(x) of each round of the game. E(x) = ______
Write a sentence that explains what this number means in terms of the game.
What is the most money a contestant could win playing the game? Find the probability this result occurs.
What is the least money a contestant could win playing the game? Find the probability this result occurs.
Should the contestant walk away at the beginning with $100 or play for a certain number of rounds? Justify your answer using the information you have discovered.