<< Subject: Probability question...
From: "J-K" ***@gci.net
Date: Wed, Apr 28, 2004 11:27
Message-id: <CATjc.14186104$***@news.easynews.com> >>
J-K - Here's the plan of attack: We'll find the probability that no ace hits
the five card board and then find what we need to add to it to equal one. Do
you need an explanation for this?
Let's call the probability that no ace hits the board Q.
Let's call the probability that at least one ace hits the board P. Do you need
an explanation for this?
The sum of these probabilities is 1. Do you need an explanation for this?
Then P + Q = 1
is the equation for "The sum of these probabilities is 1." Do you need an
explanation for this?
Let's proceed to finding Q.
There are 50 missing cards, of which four are aces. From the viewpoint of the
player with the two kings, any five of the 50 cards could make up the five card
board. The number of different five card combinations possible in 50 cards is
given by the expression C(50,5). Do you need an explanation for this?
Another way to write C(50,5) is "50 choose 5."
Another way to write C(50,5) is 50!/5!/45!
Another way to write C(50,5) is
50*49*48*47*46/1/2/3/4/5.
You're probably more familiar with writing this as
(50*49*48*47*46)/(1*2*3*4*5).
Solving, C(50,5) = 2118760.
211870 is the number of possible five card boards when you hold two kings.
Now we need to find the number of possible five card boards that do not have
any aces when you hold two kings.
If there are to be no aces, then there are only 46 cards that can be on the
five card board. Do you need an explanation for this?
The number of different five card combinations with no aces is given by the
expression C(46,5). Do you need an explanation for this?
Solving, C(46,5) = 1370754.
The probability of no aces is C(46,5)/C(50,5) =
1370754/211870. Do you need an explanation for why we're dividing here?
If not, the result of the calculation is 0.647.
Then, from P+Q = 1, doing a little algebra,
P + 0.647 = 1
P = 1 - 0.647
P = 0.353, which is the probability we seek.
You ask <<"what are the chances">>
Assuming "chances" mean "probability," the answer is 0.353.
If you want odds, they are 647 to 353 against, or
1.83 to 1 against, or
close to 2 to 1 against. Do you need an explanation for this?
1.83 to 1 is actually a bit less than 2 to 1. If you're getting at least two to
one for your money, then you have a good bet (favorable odds).
I'm not a mathematician. (But I do like to solve math problems that relate to
things that interest me). If you need any explanations along the way, perhaps
they might be better provided by one of the excellent mathematicians who
sometimes post here.
Lastly, in the famous words of Jerrod Ankenman, who took me apart in the 2004
ESCARGOT Chowaha contest (I was the chip leader with three of us left),
remember
"Math is hard."
Buzz