This is an automated reply from the desk of TA. He is sorry he can not be with you this AM because he is walking in the warm rain, to return to a log fire with the aroma of new bread and coffee in the air. He will return after therapy...
Shari:
GOOOOOOOOO
go long.
Q. Suppose you have two five-card poker hands dealt from separate decks. You are told hand A contains at least one ace. You are told hand B contains the ace of spades. Which hand is more likely to contain at least one more ace?
This (in my humble opinion) is the most difficult problem ever poised. Two hands, two ace's, two decks. Why should the odds be different? - Don't play cards with Mark.
Mark:
POKER HANDS
Hand A will have another ace approximately 12.21849% of the time.
Hand B will have another ace approximately 22.13685% of the time.
Hand B.
The following table shows the probability of 0 to 5 aces in a totally random hand.
Code:
Ace Probabilities - Random Hand
Aces Formula Combinations Probability
0 Combin(48,5) 1712304 0.658842
1 combin(4,1)*combin(48,4) 778320 0.299474
2 combin(4,2)*combin(48,3) 103776 0.03993
3 combin(4,3)*combin(48,2) 4512 0.001736
4 combin(4,4)*combin(48,1) 48 0.000018
Total 2598960 1
Take the sum for 1 to 4 aces we see the probability of at least one ace is 0.341158. The probability of two or more aces is 0.041684.
The probability of there being at least one more ace given there is at least one can be restated per Bayes' theorem as probability(two more aces given at least one ace) = probability (two or more aces)/probability(at least one ace) = 0.041684/ 0.341158 = 0.122185.
For those rusty on Bayes' Theorem ( I'm talking to you Shari) states the probability of A given B equals the probability of A and B divided by the probability of B, or Pr(A given B) = Pr(A and B)/Pr(B).
The next table shows the combinations and probability for each number of other aces given that the ace of spades was removed from the deck.
Ace Probabilities - Ace Removed Hand
Code:
Aces Formula Combinations Probability
0 combin(3,0)*combin(48,4) 194580 0.778631
1 combin(3,1)*combin(48,3) 51888 0.207635
2 combin(3,2)*combin(48,2) 3384 0.013541
3 combin(3,3)*combin(48,1) 48 0.000192
Total 249900 1
This shows the probability of at least one more ace is 0.221369.
However suppose you want to solve it the same way as hand A, using Bayes' Theorem. The probability of at least one additional ace, given the hand contains the ace of spades, could be rewritten as probability(at least two aces given ace of spades is in hand). According to Bayes' theorem this equals Probability(hand contains ace of spades and at least one more ace) / Pr(hand contains ace of spades). We can break up the numerator as Probability(2 aces including ace of spades) + Probability(3 aces including ace of spades) + Probability(4 aces). Using the first table this equals 0.039930*(2/4) + 0.001736*(3/4) + 0.000018 = 0.021285. The probability of the ace of spades is 5/52 = 0.096154.
So the probability of at least two aces given the aces of spades is 0.021285/0.096154 = 0.221369.
So the probability of two or more aces given at least one ace is 12.22% and given the ace of spades is 22.14%.
Q: Okay, I believe your numbers but it still doesn't make sense to me. I should think the probabilities would be equal. What difference does the suit make of the one ace you are given?
Let's look at another simpler situation. Suppose woman A says "I have two kids at least one is a boy." Woman B says "I have two kids and one is named John. We can assume nobody named John is a girl and no woman gives the same name to more than one kid. Using conditional probability the probability of both kids being a boy of woman A is pr(both boys)/pr(at least one boy) = pr(both boys)/(1-pr(both girls)) = (1/4)/(1-(1/4)) = (1/4)/(3/4) = 1/3.
However the probability woman B's other kid is a boy, or both kids are boys, is ½, because saying one kid is named John tells us nothing about the other kid.
To make another example suppose you go to Jiffy Lube and they offer two deals for the same price. Deal A is they will check four parts and replace only the first defective one found. Deal B is they will check only one problem and fix it if found. Wouldn't you rather take deal A? Your car came in with the same number of expected bad parts but the probability of finding a problem is greater under deal A, and thus you will leave with a small number of expected defective parts under that plan. Likewise a test for any ace will probably turn up the only ace, while a test for the ace of spades does not check for the other three suits, leaving them more likely to be aces.
Mark, I acknowledge the poke in the eye with a sharp stick. I am only grateful I did not violate rule 5. Boy, there is no comeback from that.
Shari, sorry I missed your fishing question. I am running a programme for the University of California at Berkeley on climate change, and it shut down all other programmes. I think it was Mark's revenge.
BTW I do fish, but get this; I do not use a hook. Saves the fish the pain, and does not spoil the tranquillity of the moment.
Mark, your photograph idea is ace. Providing it is taken in the best possible taste. I think I have on old Kodak box brownie No 2 around here somewhere. Although if the truth be known, the box is worth many times the value of the camera.
"Staying up late, eh Try?"
My mom always goes to the Roxy on a Thursdays, so I get to stay up listening to the Eagles:
One Of These Nights
One of these nights
One of these crazy old nights
We're gonna find out
Pretty mama
What turns on your lights
The full moon is calling
The fever is high
And the wicked wind whispers
And moans
You got your demons
You got desires
Well, I got a few of my own
Oo, someone to be kind to in
Between the dark and the light
Oo, coming right behind you
Swear I'm gonna find you
One of these nights
One of these dreams
One of these lost and lonely dreams
We're gonna find one
One that really screams
I've been searching for the daughter
Of the devil himself
I've been searching for an angel in white
I've been waiting for a woman who's a little
Of both
And I can feel her but she's nowhere
In sight
Oo, loneliness will blind you
In between the wrong and the right
Oo, coming right behind you
Swear I'm gonna find you
One of these nights
One of these nights
In between the dark and the light
Coming right behind you
Swear I'm gonna find you
Get 'ya baby one of these nights
One of these nights
One of these nights
I can feel it
I can feel it
One of these nights
Coming right behind you
Swear I'm gonna find you now
One of these nights..
B
BA
BACK
DOX
DOX
the woKINGods
There is a one acre field in the shape of a right triangle, with sides of length x and y. At the midpoint of each side, there is a post. Tethered to the posts on each side is a sheep.
Tethered to the post on the hypotenuse is a dog. Each animal has a rope just long enough to reach the two adjacent vertices of the triangle.
How much area outside of the field do the sheep have to themselves
Missed the riddle: