A Standard Deck of Cards Has 52 Cards As Described Below
Math 125
Investigation #5
Investigation #5: “Raise or Fold?”
A standard deck of cards has 52 cards as described below:
-26 red & 26 black
-13 heart, diamonds, spades and clubs.
-4 of every number 1 – 10, Jacks, Queens, Kings and Aces
Winning hands in poker are as followed – a pair being the worst and a Royal Flush being the best.
A pair - 2 of same number 2 – 10, Jack, Queen, King or Ace.
Two pair - 2 of one number 2 – 10, J, Q, K or A and 2 of another.
3 of a kind – 3 of the same card 2 – 10, J, Q, K or A.
Straight – 5 consecutive cards NOT in the same suit
Flush – all the same suit
Full House – a pair and 3 of a kind (Ex. 2, 2, 9, 9, 9)
Straight Flush – 5 consecutive cards of the same suit (Ex. 3, 4, 5, 6, 7 of spades)
Royal Flush – A, K, Q, J, 10 of all the same suit
Part I (5pts):
Calculate the probability of being dealt ONLY 1 of the hands above. FYI – in the case of the first 3 hands it does NOT matter what the remaining cards in your hand are. You may research the calculations for these probabilities, but you MUST include an explanation of the calculations. For example, you can find the probabilities on Wikipedia, but the explanations are not very clear.
Scoring:
Calculation: 1 pt.
Explanation: 4pts
Ex. The probability of a pair is 42.3%.
Calculation:
Explanation:
13 C 1 is because of the 13 face values you want 2 of them to be the same. So to start it does not matter which card you are dealt – it can be any 1 of the 13 face values
4 C 2 – regardless of which face value you were dealt for your first card, there are 4 of them altogether – you want 2 of those 4.
12 C 3 is because there are 12 face values remaining that are different from the first 2 cards you were dealt. For your remaining 3 cards you want 3 different face values.
4 C 1 is because you want only 1 of the 12 remaining face values. This is cubed because you need3 different face values for your remaining 3 cards. Otherwise you’d have 2 pair or something else.
*** Since a pair was given as an example, you must choose one of the other hands for this assignment, but you only have to do 1.
Part II: Jack’s or Better (5pts)
In 5 – card draw each player is dealt 5 cards and then given the opportunity to discard up to 3 cards for new ones. For each scenario below use probability to decide which cards you would discard that would give you the best chances of getting a pair of Jacks or better - i.e any pair Jacks or higher or any hand higher than a pair. All answers should include the following:
-What you plan to discard (1pt.)
-What card(s) you could be dealt to get a winning hand (2pt.)
-The probability you will get the card(s) you want (2pts.)
Scenario #1:
Your hand:
9K47Q
Scenario #2:
Your hand:
44K107
Scenario #3:
Your hand:
3 910JQ
Part II Answer:
Scenario #1:
- Discard 4 of hearts, 7 of hearts, and 9 of spades -Keep K of diamonds and Q of hearts.
Winning drawProbability
a. At least 1 K or Q / 34.26%b. At least 2 Aces or Jacks / 3.23%
c. 3 of kind: 2,3,5,6,8,10 / 0.15%
d. 3 of kind: 4, 7 or 9 / 0.02%
e. Straight / 0.79%
total / 38.45%
Discarding the three low cards and keeping the Q and K is the best option. Keeping a third card (e.g. the 9) does not improve the odds of getting Jacks or better. Other options, such as discarding the K and 9 to get a heart flush, produce even worse odds.
Scenario #2:
- Discard10 of hearts, and 7 of clubs -Keep K of spades, 4 of clubs, and 4 of diamonds.
Winning drawProbability
a. At least 1 K / 12.49%b. At least one 4 / 8.42%
c. Pair: J,Q,Aces / 1.67%
total / 22.57%
In this case, discarding two cards is better. Keeping just the pair of fours does not give the best odds. Instead, keeping the K as the third card gives the best odds. Keeping two cards, e.g. K, 10, would not be as beneficial.
Scenario #3:
- Discard3 of spades, 9 of spades, and 10 of hearts -KeepQ of clubs, and J of spades.
Winning drawProbability
a. At least 1 J or Q / 34.26%b. At least 2 Aces or Jacks / 3.23%
c. 3 of kind: 2,4,5,6,7,8 / 0.15%
d. 3 of kind: 3, 9 or T / 0.02%
e. Straight / 0.79%
total / 38.45%
Getting Jacks or better by discarding 3 and keeping two face cards are the best odds.