Blackjack, or 21, is a popular casino game that begins with each player and the dealer...

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 player counts the ace as 1 or 11 points. The goal is to obtain a hand with a value as close as possible to 21 without exceeding 21. After the initial deal, each player and the dealer may draw additional cards (called “taking a hit”) in order to improve her or his hand. If a player or the dealer takes a hit and the value of the 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 decides whether to take a hit. 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 (face showing) and one card down (face hidden). The player then decides whether to take a hit based on knowledge of the dealer’s up card. Suppose that you are playing blackjack and the dealer’s up card is a 6 and your hand has a value of 16 for the two cards initially dealt.

With a hand of a value of 16, if you decide to take a hit, the following cards will improve your hand: ace, 2, 3, 4, or 5. Any card with a point count greater than 5 will result in you going broke. Assume that if you have a hand with a value of 16, the following probabilities describe the ending value of your hand:

Value of Hand	17	18	19	20	21	Broke
Probability	0.0769	0.0769	0.0769	0.0769	0.0769	0.6155

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:

Value of Hand	17	18	19	20	21	Broke
Probability	0.1654	0.1063	0.1063	0.1017	0.0972	0.4231

Construct a simulation model to simulate the result of 1,000 blackjack hands when the dealer has a 6 up and you take a hit with a hand that has a value of 16. What is the probability of the dealer winning, a push (a tie), and you winning, respectively?
If you have a hand with a value of 16 and don’t take a hit, the only way that you can win is if the dealer goes broke. If you don’t take a hit, what is the probability of the dealer winning, a push (a tie), and you winning, respectively?
Based on the results from parts (a) and (b), should you take a hit or not if you have a hand of value 16 and the dealer has a 6 up?

Expert Solution

orchestra answered 2 years ago

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