Blackjack, or 21, is a popular casino game that begins with each player and the dealer being dealt two cards. The value of each hand is determined by the point total of the cards in the hand. Face cards and 10s count 10 points, aces can be counted as either 1 or 11 points, and all other cards count at their face value. For instance, the value of a hand consisting of a jack and an 8 is 18; the value of a hand consisting of an ace and a two is either 3 or 13 depending on whether the ace is counted as 1 or 11 points. The goal is to obtain a hand with a value of 21, or as close to it as possible without exceeding 21. After the initial deal, each player and the dealer may draw additional cards (called “taking a hit”) in order to improve their hand. If a player or the dealer takes a hit and the value of their hand exceeds 21, that person “goes broke” and loses. The dealer’s advantage is that each player must decide whether to take a hit before the dealer. If a player takes a hit and goes over 21, the player loses even if the dealer later takes a hit and goes over 21. For this reason, players will often decide not to take a hit when the value of their hand is 12 or greater. The dealer’s hand is dealt with one card up and one card down. The player then decides whether to take a hit based on knowledge of the dealer’s up card. A gambling professional determined that when the dealer’s up card is a 6, the following probabilities describe the ending value of the dealer’s hand:

a. Set up intervals of random numbers that can be used to simulate the ending value of the dealer’s hand when the dealer has a 6 as the up card.
b. Use the random numbers in column 4 of Table 16.2 to simulate the ending value of the dealer’s hand for 20 plays of the game.
c. Suppose you are playing blackjack and your hand has a value of 16 for the two cards initially dealt. If you decide to take a hit, the following cards will improve your hand: ace, 2, 3, 4, and 5. Any card with a point count greater than 5 will result in you going broke. Suppose you have a hand with a value of 16 and decide to take a hit. The following probabilities describe the ending value of your hand:

Use the random numbers in column 5 of Table 16.2 to simulate the ending value of your hand after taking a hit for 20 plays of the game.
d. Use the results of parts (b) and (c) to simulate the result of 20 blackjack hands when the dealer has a 6 up and the player chooses to take a hit with a hand that has a value of 16. How many hands result in the dealer winning, a push (a tie), and the player winning.
e. If the player has a hand with a value of 16 and doesn’t take a hit, the only way the player can win is if the dealer goes broke. How many of the hands in part (b) result in the player winning without taking a hit? On the basis of this result and the results in part (d), would you recommend the player take a hit if the player has a hand with a value of 16 and the dealer has a 6up?

Transcribed Image Text:

Value of Ha17 18 19 20 2 Broke Probability0.1654 0.1063 0.1063 0.1017 0.0972 0.4231 Value of Hand7 19 20Broke Probability 0.0769 0.0769 0.0769 0.0769 0.0769 0.6155