Calculating Chances of Descending Order with 36 Cards

In summary, the conversation discusses the chances of four randomly dealt cards being in descending order from a set of 36 cards. The participants go through different approaches and come up with the answer of 1/(34*35*36) for the probability of consecutively descending cards, and 1/8 for non-consecutive descending cards.
  • #1
stupidkid
18
0
The cards in a set of 36 are numbered 1 to 36. The cards are shuffled and four cards are dealt. What are the chances of them being dealt in descending order?
Major help needed
I have tried everything in this problem my answer was 1/8 which was wrong I hope you guys can do better.
 
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  • #2
Well how many different sets of 4 cards can you get that are in descending order? (i.e. {36, 35, 34, 33} is one, {35, 34, 33, 32} is another... how many are there?)
 
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  • #3
either the second is higher than the first, or the third is higher than on of the previous two or the 4 is higher than one of the previous 3. hmm, A or B or C... inc exc principle then a bit of coditional prob. anyone?
 
  • #4
It's actually fairly simple. First you can easily determine how many different sets of 4 cards can you get that are in descending order, and you can probably figure out how many total hands there are. And well, yea... you know what to do from there.
 
  • #5
but there can 1000000000000000s of possibilties for eg 7,6,5,1 or 10,6,3,2
like that
 
  • #6
stupidkid said:
but there can 1000000000000000s of possibilties for eg 7,6,5,1 or 10,6,3,2
like that

Actually, your estimate is much larger than the actual number of possibilities. Try to think of it this way:

How many possibilities are there for the first card? (Ans: Obviously 36)
How many possibilities are there for the second card? (Hint: Not 36)
How many possibilities are there for the third card?
How many possibilities are there for the fourth card?

Now you can multiply them up and it gives you the total number of possibilities for a 4-card deal.
 
  • #7
its not as easy as that.
 
  • #8
given 4 distinct cards, a,b,c,d, each hand containing these 4 is equally likely to be dealt, exactly one of these is descending. how many ways of dealing a,b,c and d are there ie how many ways can you reciecve a hand with the cards 1,2,3,4 in? how many diffeent hands are there in total?
 
  • #9
I thought you meant consecutively descending. Ok, so the way I thought about it was this:

You can add the number of ways you can get a descending order where your highest card (call it M) is 4 (since M can't be less than 4), plus the number of possibilities when M=5, M=6, all the way to M=36.

The way of counting the number of ways to get a descending order when your highest card is M is (at least I think) easiest to see in the form of code:

possible_descending_hands = 0;
for (i=3, i < M, i = i + 1){
for (j=2, j < i, j = j + 1){
for (k=1, k < j, k = k + 1){
possible_descending_hands = possible_descending_hands + 1;
}
}
}

Then of course, you loop again for M = [4,36] and that would be your number of possible descending hands. Note that I'm not suggesting you write a program for this. It's just that it's easiest to explain how to count your descending hands with the use of loops (you can yet convert that into a mathematical expression).

[Edit: Hmm.. not allowed to put spaces at the beginning of lines. I guess that means no indentation for you :P]
 
  • #10
well, you could do that but the answer is can be obtained just from thinking about it if it's any help the extreme cases are

1. dealing 1 card in descending order: 1/1

2. Dealing 36 cards in descending order 1/36!

and i thought i gave sufficient hints last time!
 
  • #11
For the case of consecetively decending it's also very easy.

The first card must be 4 or more which has a probabilty of 33/36. After that there is no choice which card must come next so the probability is just 33/36 * 1/35 * 1/34 * 1/33 = 1/(34*35*36).
 
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  • #12
the question doesn't say they are in sequence.
 

1. What is the likelihood of a specific order of 36 cards appearing?

The likelihood of any specific order of 36 cards appearing is very low, as there are over 8x10^39 possible combinations of the 36 cards.

2. How is the chance of descending order calculated?

The chance of descending order is calculated by taking the number of possible combinations that result in descending order (1) and dividing it by the total number of possible combinations (8x10^39).

3. Is there a formula to calculate the chances of descending order with 36 cards?

Yes, the formula for calculating the chances of descending order with 36 cards is 1/8x10^39.

4. Can the chances of descending order be increased by shuffling the cards?

No, shuffling the cards does not change the total number of possible combinations and therefore does not affect the chances of descending order.

5. Are there any factors that can influence the chances of descending order?

The chances of descending order are solely dependent on the number of cards (36) and the total number of possible combinations (8x10^39), so there are no external factors that can influence it.

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