Card Drawing Probabilities
- Thread starter Ronin Storm
- Start date
Huung
Well-Known Member
It is indeed 6 factorial, so 6*5*4*3*2*1
Also, just to clarify a few things...
In your original post you said a deck of 10, with three draws, looking for the probability of drawing a 10.
Would you stop drawing if you found the 10 on the first or second draw, or would it be a blind draw of 3 cards?
If you were to put the cards back after each attempt, it would be pretty simple.
Code:
1 - (chance of failing to get desired card ^ number of attempts)So in this case, if you put back the cards each time, you'd be looking at:
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1 - (0.9 ^ 3) = 0.271 = 27.1%If you were to keep the card out, you do similar to what Haven initially said:
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1 - (9/10 * 8/9 * 7/8) = 0.3 = 30%Which makes sense, you have 3 cards out of 10, so you'd expect your chance that one of those cards is the 10 to be 3 out of 10 (you've had '3 out of 10' chances to pick it out).
For the problem I'm dealing with, it's a case of drawing without replacement (within a sequence of draws). Thus it's your last example.
The problem I'm assessing is this:
I have a board with numbered spawn points on the board; let's say there are five spawn points. Each turn, aliens can spawn at those spawn points. The points they spawn at are chosen by cards drawn from a prepared deck. Thus, in this case, I might have a deck of 10 cards numbered 1 to 5 with two instances of each.
In this particular case, the chances of drawing a single alien over two draws (assuming two spawns a turn) is:
Although, there's another oddity that I haven't accounted for... what if the first draw is actually a "success":
This is the chances of drawing a second alien after drawing a first, but that doesn't tally for me. Is this actually the chance of drawing two aliens at the same point?
The problem I'm assessing is this:
I have a board with numbered spawn points on the board; let's say there are five spawn points. Each turn, aliens can spawn at those spawn points. The points they spawn at are chosen by cards drawn from a prepared deck. Thus, in this case, I might have a deck of 10 cards numbered 1 to 5 with two instances of each.
In this particular case, the chances of drawing a single alien over two draws (assuming two spawns a turn) is:
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1 - ( ( 8 * 7 ) / ( 10 * 9 ) ) = 1 - ( 56 / 90 ) = 0.378 = 37.8%Although, there's another oddity that I haven't accounted for... what if the first draw is actually a "success":
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1 - ( ( 8 * 8 ) / ( 10 * 9 ) ) = 1 - ( 64 / 90 ) = 0.289 = 28.9%This is the chances of drawing a second alien after drawing a first, but that doesn't tally for me. Is this actually the chance of drawing two aliens at the same point?
Huung
Well-Known Member
If you draw an alien in the first turn, you then only have a 1/9 chance of drawing one in the second. Therefore the chance of drawing both instances of the same exact alien in two draws would be a straight multiplication of the two chances:
As there are 5 different types of aliens, there are 5 ways in which this can occur. Therefore you just multiply this up by 5 times to get 11.11% (1/9).
To find two at exactly the same time, you'd have a 1/5 chance, followed by a 1/5 chance (as you'd assume no card was actually taken). In this case you're looking at a 1/25 * 5 chance, or 1/5 chance of getting a matching pair of aliens. Which still doesn't match the number you have :\
I'll try and figure out how you got that number, and get back to you!
Code:
1/5 * 1/9 = 0.022 = 2.22%As there are 5 different types of aliens, there are 5 ways in which this can occur. Therefore you just multiply this up by 5 times to get 11.11% (1/9).
To find two at exactly the same time, you'd have a 1/5 chance, followed by a 1/5 chance (as you'd assume no card was actually taken). In this case you're looking at a 1/25 * 5 chance, or 1/5 chance of getting a matching pair of aliens. Which still doesn't match the number you have :\
I'll try and figure out how you got that number, and get back to you!
Huung
Well-Known Member
This is the image I have in my head (excuse the MS Paint-ness!).
Is this something like what you have intended? The spawn points at the top, numbered 1 through 5, and the 10 cards, with doubles of each type.
I have been assuming all along that every card has an alien of some type on it, and that no cards are "blanks".
In this case, say you're looking for the green alien (card 1).
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Green + !Green
1/5 * 8/9 = 8/45
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!Green + Green
4/5 * 2/9 = 8/45
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Green + Green
1/5 * 1/9 = 1/45
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Any Greens
8/45 + 8/45 + 1/45 = 17/45 = 0.378 = 37.8%This is the value you had for "drawing a single alien over two draws".
I'm not totally sure why you'd need the second value, as a first draw 'success' (by which I take it you'd find a 'Green') doesn't affect the overall outcome.
Awesome aliens. My system is slightly different, but I'll come back to that as functionally the maths works out the same.
Right, I'm glad that you and I get to the same number but it interests me that the way you've done it describes clearly what the result means. In this case, the chances of getting at least one alien (but maybe two) at a particular point is 37.8%.
The thing I'm having trouble with is figuring out what is meant by the second number...
That said, the value of 37.8% is good enough for what I'm needing. I'm looking at the odds of anything spawning, in a given turn, at a particular spawn point. Here's why...
Mission: Hack the ship's computer.
Phase 1: Marines move through the ship, fighting off aliens, to get to the command centre. When they start the hack, begin phase 2.
Phase 2: Marines are attacked in heavy waves while hacking. When the hack completes, begin phase 3.
Phase 3: Marines evacuate through a nearby door they've opened by hacking.
In phase 1, aliens appear at any of 4 spawn points (1-4), but points 2 and 3 are ones they prefer. Thus, the spawn deck contains 1, 2, 2, 3, 3, 4 (6 cards).
In phase 2, aliens stop appearing at points 1 and 4, slow down in 2 and 3, but now spawn at 5, 6 and 7. New distribution of 2, 3, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7 (12 cards).
In phase 3, the manic spawn at points 6 and 7 dies down but aliens are still coming from all over. Distribution of 2, 2, 3, 3, 5, 5, 6, 6, 7, 7 (10 cards).
What I want to know is for a given spawn point across the phases, how does its usage potential change. I assess this based on the likelihood of at least a single alien spawning at that point. Clearly, in the examples above, there are fluctuations. The interesting thing is that the fluctations aren't as wild as they at first feel.
Right, I'm glad that you and I get to the same number but it interests me that the way you've done it describes clearly what the result means. In this case, the chances of getting at least one alien (but maybe two) at a particular point is 37.8%.
The thing I'm having trouble with is figuring out what is meant by the second number...
That said, the value of 37.8% is good enough for what I'm needing. I'm looking at the odds of anything spawning, in a given turn, at a particular spawn point. Here's why...
Mission: Hack the ship's computer.
Phase 1: Marines move through the ship, fighting off aliens, to get to the command centre. When they start the hack, begin phase 2.
Phase 2: Marines are attacked in heavy waves while hacking. When the hack completes, begin phase 3.
Phase 3: Marines evacuate through a nearby door they've opened by hacking.
In phase 1, aliens appear at any of 4 spawn points (1-4), but points 2 and 3 are ones they prefer. Thus, the spawn deck contains 1, 2, 2, 3, 3, 4 (6 cards).
In phase 2, aliens stop appearing at points 1 and 4, slow down in 2 and 3, but now spawn at 5, 6 and 7. New distribution of 2, 3, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7 (12 cards).
In phase 3, the manic spawn at points 6 and 7 dies down but aliens are still coming from all over. Distribution of 2, 2, 3, 3, 5, 5, 6, 6, 7, 7 (10 cards).
What I want to know is for a given spawn point across the phases, how does its usage potential change. I assess this based on the likelihood of at least a single alien spawning at that point. Clearly, in the examples above, there are fluctuations. The interesting thing is that the fluctations aren't as wild as they at first feel.
Huung
Well-Known Member
Ah I see. So you're looking at the spawn points being the "cards" then, as opposed to the variety of alien?
So, say 'x' aliens can spawn per "turn", and a phase consists of 'y' turns, you want to know which spawn points they're most likely to appear at?
Assuming, as you said, a single alien spawning, the above table would stay as is.
A further column could be added, as a cumulative chance of any alien appearing at that spawn over the course of all three phases. Assuming again a single alien spawn per phase, the table would be:
I converted them all into 60ths to make the mental maths easier. The 180 is due to 3 aliens being picked overall (60 * 3 = 180). This 'Usage %' column would hold true for any number of aliens per phase, provided each phase spawned the same number. Obviously if the second phase is suddenly spawning far greater numbers, the weighting for the points will change.
So, say 'x' aliens can spawn per "turn", and a phase consists of 'y' turns, you want to know which spawn points they're most likely to appear at?
Code:
Point | Phase 1 | Phase 2 | Phase 3
1 1/6 - -
2 1/3 1/12 1/5
3 1/3 1/12 1/5
4 1/6 - -
5 - 1/6 1/5
6 - 1/3 1/5
7 - 1/3 1/5Assuming, as you said, a single alien spawning, the above table would stay as is.
A further column could be added, as a cumulative chance of any alien appearing at that spawn over the course of all three phases. Assuming again a single alien spawn per phase, the table would be:
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Point | Phase 1 | Phase 2 | Phase 3 | Any usage | Usage %
1 10/60 - - 10/180 5.56%
2 20/60 5/60 12/60 37/180 20.56%
3 20/60 5/60 12/60 37/180 20.56%
4 10/60 - - 10/180 5.56%
5 - 10/60 12/60 22/180 12.22%
6 - 20/60 12/60 32/180 17.78%
7 - 20/60 12/60 32/180 17.78%I converted them all into 60ths to make the mental maths easier. The 180 is due to 3 aliens being picked overall (60 * 3 = 180). This 'Usage %' column would hold true for any number of aliens per phase, provided each phase spawned the same number. Obviously if the second phase is suddenly spawning far greater numbers, the weighting for the points will change.