Page 202 - DMTH202_BASIC_MATHEMATICS_II

Unit 14: Probability

This entirely mention P. Particularly, we can locate the probability that the point occurs into any Notes
(sufficiently nice) set A as the length of that set.
Some examples of probability are discussed below.

Example: A Poker Hand
The game of poker has various variants. General to all is the truth that players obtain - one way
or another - hands of five cards each. The hands are evaluated according to a predestined ranking
system. Now we shall assess probabilities of numerous hand combinations.
Poker utilizes the standard deck of 52 cards. There are C(52, 5) probable combinations of 5 cards
chosen from a deck of 52: 52 cards to select the first of the five from, 51 cards to select the second
one, ..., 48 to select the fifth card. The product 52 × 51 × 50 × 49 × 48 must be divided by 5!since the
order in which the five cards are added to the hand is of no significance, such as 78910J
is the same hand as 9710J8. So there are C(52, 5) = 2598960 different hands. The poker
sample space comprises 2598960 uniformly probable elementary events.
The probability of either hand is obviously 1/2598960. Visualize having an urn with 52 balls, of
which 5 are black and the remaining white. You are to draw 5 balls out of the urn. What is the
probability that all 5 balls drawn are black?
The probability that the first ball is black is 5/52. Presuming that the first ball was black, the
probability that the second is also black is 4/51. Presuming that the first two balls are black, the
probability that the third is black is 3/50, ... The fifth ball is black with the probability of 1/48,
given the first 4 balls were all black. The probability of drawing 5 black balls is the product:

3 2 1 1
  
50 49 48 C 52,5 
The highest ranking poker hand is a Royal Flush - a series of cards of the same suit beginning
with 10, e.g., 10JQKA. There are 4 of them, one for each of the four suits. So the probability
of getting a royal flush is 4/2598960 = 1/649740. The probability of getting a royal flush of, say,
spades , is obviously 1/2598960.
Any sequence of 5 cards of the same suit is a straight flush ranked by the highest card in the
sequence. A straight flush may begin with any of 2, 3, 4, 5, 6, 7, 8, 9, 10 cards and some times with
an Ace where it is thought to have the rank of 1. So there are 9 (or 10) possibilities of getting a
straight flush of a specified suit and 36 (or 40) possibilities of obtaining any straight flush.
Five cards of the same suit - not essentially in sequence - is a flush. There are 13 cards in a suit and
C(13, 5) = 1287 combinations of 5 cards out of 13. All in all, there are 4 times as many flush
combinations: 5148.
Four of a kind is a hand, such as 5555K, with four cards of the similar rank and one extra,
unmatched card. There are 13 combinations of 4 equally ranked cards each of which can complete
a hand with any of the remaining 48 cards. A hand with 3 cards of one rank and 2 cards of a
dissimilar rank is called Full House. For a specified rank, there are C(4, 3) = 4 methods to select 3
cards of that rank; there 13 ranks to consider. There are C(4, 2) = 6 combinations of 2 cards of equal
rank, but now only 12 ranks to select from. There are then 4 × 13 × 6 × 12 = 3744 full houses.
A straight hand is a straight flush without “flush”, so to articulate. The card must be in series but
not essentially of the same suit. If the ace is permitted to begin a hand, there are 40 ways to select
the first card and then, we need to account that the remaining 4 cards could be of any of the 4
suits, providing the total of 40 × 4 × 4 × 4 × 4 = 10240 hands. Removing 40 straight flushes leaves
10200 “regular” flushes.

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