. An opening hand in the game Blackjack is made up of two cards with the following point values:
• Aces are worth 11 points.
• Jacks, Queens, and Kings are worth 10 points each.
• All other cards are worth their number (ie a 4 is worth 4 points, a 7 is worth 7 points, etc.)
Greatly oversimplifying things, the goal of the game is to get exactly 21 points. In the questions
below, assume that an opening hand is drawn from a freshly shuffled deck, and that cards are put
back and then reshuffled in between trials. (Note: this is not how blackjack actually works.) For
this question, show all of your work.
(A) What is the expected value of an opening hand of blackjack?
(B) What is the variance of an opening hand of blackjack?
(C) What is the probability of getting exactly 21 in an opening hand?
(D) If you know that the first card you draw is an ace, what is the probability of getting exactly
21 in an opening hand?
Answers
Probability of getting exactly 21 in an opening hand = 0.048
Step-by-step explanation:
Ace = 11
Jack , Queen , kings , & card 10 = 10
To get 21 we need to get
Ace + either of Jack , Queen , kings , & card 10
Ace = 4
either of Jack , Queen , kings , & card 10 = 4 * 4 = 16
probability of getting exactly 21 in an opening hand
= (4/52)(16/51) + (16/52)(4/51)
= 2 * 64 / (52 * 51)
= 64/ (26 * 51)
= 32/(13 * 51)
= 32/663
= 0.048
if first card you drawn is an ace
then probability of getting exactly 21 in an opening hand = 16/51
= 0.314
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