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WRITTEN ASSIGNMENT 3
CASE STUDY: TEXAS HOLD'EM
STA-201: PRINCIPLES OF STATISTICS
a. The probability that you are dealt pocket aces is 1/221, or 0.00452 to three significant digits. If you studied either Section 4.5 and 4.6 or Section 4.8, verify that probability.
1st Card- 4 cards that are aces out of the 52 cards in the deck; so 4/52 = 0.0769 or slightly less than an 8% chance that the first card is an Ace.
2nd Card- Since first card dealt was an Ace, there are only 3 aces out of 51 card remaining; so 3/51 = 0.0588 or slightly less than a 6% chance that a second ace will be dealt.
Therefore to find the probability that these two events will happen, you will need to take the results of both and multiply them together. 0.0769 x 0.0588= 0.0452172 rounded to three significant digits is 0.00452 is the probability of getting dealt pocket aces. So the probability of 1 in 221 is correct.
b. Using the result from part (a), obtain the probability that you are dealt 'pocket kings.'
Same probability as part (a).
I find that it is unnecessary to repeat the exact same calculations since the result will be the same as part (a.). However, since the problem is quite vague, I decided that calculating the result of pocket kings after pocket aces are drawn, is more fitting.
1st Card Ace- 4 cards that are aces out of the 52 cards in the deck; so 4/52 = 0.0769 or slightly less than an 8% chance that the first card is an Ace.
2nd Card Ace- Since first card dealt was an Ace, there are only 3 aces out of 51 card remaining; so 3/51 = 0.0588 or slightly less than a 6% chance that a second ace will be dealt.
0.00452 is the probability of getting dealt pocket aces (or pocket kings or any pocket pair for that matter).
3rd Card King- 4 cards out of 50 cards left in the deck; 4/50= 0.08 or an 8% chance to draw a king.
4th Card King- Only 3 kings out of 49 cards; 3/49= 0.0612 or slightly more than a 6% chance to draw the second king after two aces are drawn.
Therefore: 0.0769 x 0.0588 x 0.08 x 0.0612 = 0.0000221 or a 1/45,249 chance of these happening. Ironically this happened to me when playing. I drew the pocket kings and another guy drew the pocket aces. After the flop, I went all in, since there was a king in the flop. Only two guys called and one had pocket Aces. Damn if he didn’t get an Ace on the river card. Cost me $100. Quit playing Hold-em after that.
c. Using the result from part (a) and your analysis in part (b), find the probability that you are dealt a 'pocket pair,' that is, two cards of the same denomination.
2nd Card Match- Since first card dealt has no relevance: there are only 3 aces out of 51 cards remaining; so 3/51 = 0.0588 or slightly less than a 6% chance of drawing pocket pairs.
d. contains at least 1 card of your denomination. (Hint: Complementation Rule.)
Using the Complementation Rule: P(E)=1-P(not E)
1st Card- There are 2 cards that's your denomination out of 50.
2nd Card- There are 48 cards that's not the pair denomination out of 49.
3rd Card- There are 44 cards that would not make a pair out of 48.
Since there are three cards on the flop, that means there are three possible outcomes and need to multiply the three events by three.
1-((2/50)(48/49)(44/48))x3=1-((0.04)(0.978)(0.916)x3)=1-0.108(0.10750176 actual), means that there is an 89.2% chance of trips Not happening.
e. gives you 'trips,' that is, contains exactly 1 card of your denomination and 2 other unpaired cards.
1st Card- There are 2 cards that's your denomination out of 50.
2nd Card- There are 48 cards that's not the pair denomination out of 49.
3rd Card- There are 44 cards that would not make a pair out of 48.
Since there are three cards on the flop, that means there are three possible outcomes and need to multiply the three events by three.
((2/50)(48/49)(44/48))x3=((0.04)(0.978)(0.916))x3=0.108(0.10750176 actual), means that there is a 10.8% chance of trips happening.
f. gives you “quads,” that is, contains 2 cards of your denomination
1st Card- There are 2 cards that's your denomination out of 50.
2nd Card- There are 1 card left that is your denomination out of 49.
3rd Card- There are 48 cards that would not make a pair out of 48.
Since there are three cards on the flop, that means there are three possible outcomes and need to multiply the three events by three.
((2/50)(1/49)(48/48))x3=((0.04)(0.0204)(0))x3=0.00245(0.002448 actual), means that there is a 0.25% chance of trips happening.
g. gives you a “boat,” that is, contains 1 card of your denomination and 2 cards of another denomination.
1st Card- There are 2 cards that's your denomination out of 50.
2nd Card- There are 48 cards not in your denomination out of 49.
3rd Card- There are 3 cards that would make a pair out of 48.

Texas Holdem Statistics

Texas Holdem Rules

Since there are three cards on the flop, that means there are three possible outcomes and need to multiply the three events by three.

Texas Holdem Statistic Chart

((2/50)(48/49)(3/48))x3=((0.04)(0.978)(0.0625))x3=0.00734(0.007335 actual), means that there is a 0.73% chance of trips happening. (Hmmmnnn… I would have thought this would be lower than four of a kind since four of a kind is a higher winning hand.)