Probability is something that drives me crazy. On the one hand, it makes sense. If I randomly select one card from a standard deck, I know it’s “unlikely” to be the two of hearts. Then again, if it is the two of hearts, it is still “unlikely” to be so. This is where everything goes down the toilet for me.
Yes, I realize one of the challenges of studying probability is that much of it is counterintuitive. That’s not my problem. My problem is that, often times, probability doesn’t really seem to mean anything at all. At all! For example, whilst eating M&Ms the other day, I had the following thought:
Say you have ten M&Ms in a bag. Five are red, five are blue. If you reach in and pull out any two (without looking, of course), odds are you’ll pull out two of the same color. Why? Because there are three possible outcomes: (1) two reds, (2) two blues, and (3) one of each color. This means there is a 2 in 3 chance of pulling out two of the same color.
As far as I understand, this is correct. Now here’s the problem: if you perform the same experiment but pull out one M&M at a time (even without looking until you’re completely done), your odds change. You suddenly have a 50% chance of pulling out different colors. Why? Because there are four possible outcomes. In order of pulling them out, these are the potential results:
- Red, red
- Red, blue
- Blue, red
- Blue, blue
Okay, I admit, once you pull out one red M&M, you then have a 5 in 9 chance of pulling out a blue. Therefore, by pulling them out individually, your odds of pulling out different colors increases just a tad. But not enough to jump from 33% to 50% odds! That just seems ludicrous. But isn’t that what we’re obligated to believing?
“Truth,” if there is such a thing (as I believe there is), shouldn’t be dependent upon our knowledge of it. But probability is wholly based on knowledge. If three men are in a room and we know one of them is a murderer, do they each have a 33% likelihood of being the murderer? Well, the non-murderers aren’t somehow 33% murderers, just as the actual murderer is not 33% a murderer. The murderer knows who the murderer is, so for him there is no such thing as probability in the matter. Thus, if probability is in any sense true, it is only true in a relative sense.
Things like this drive me crazy. I don’t know why it’s been in my head lately, but I’m not too fond of it. Thanks for letting me rant.
*This may not be 100% accurate, but that’s ultimately beside the point. Don’t be anal. Just read the post and understand what I’m getting at. We all accept that a coin toss very closely approximates a 50-50 scenario.
Your post got me thinking. You say:
ReplyDelete"...there are three possible outcomes: (1) two reds, (2) two blues, and (3) one of each color. This means there is a 2 in 3 chance of pulling out two of the same color."
Just because there are three possible combinations of two M&Ms when you draw two at a time (red/red, red/blue, and blue/blue) does not mean that each combination has the same likelihood.
Let's label the two M&Ms that you draw as A and B. If A is red, then that means there are four other reds in the bag, and five blues. Or, expressed in fractional terms, four of the remaining nine M&Ms are red (ie, about 44% of them), and five of the remaining nine M&Ms are blue (56%). Therefore, there is a (roughly) 44% chance of B being red, and a 56% chance of B being blue. To get the probability of a certain color combination such as red/red or red/blue, you have to multiply the probability of getting the color you want in the first slot (50%) times the probability of getting the color you want in the second slot (either 44% or 56%, depending on which color you want, and whether A is the same color as B). The problem is that by drawing two at a time, you can't tell which is A and which is B. This is corrected when you change to drawing two M&Ms in succession.
For the first M&M (A), there are five M&Ms of each color, so there is a 50% chance that A will be red, and a 50% chance that A will be blue. But on the second draw (B), the probability of drawing each color is dependent on what color A is, because there are either four reds and five blues left in the bag, or five reds and four blues. Remember that to get the correct probability of each combination, you have to multiply the probability of getting each A times the probability of getting each B. So the possible outcomes, with probabilities, are as follows:
1 - A is red (50% chance), B is red (44% chance). 50% * 44% = 22% chance of drawing this set.
2 - A is red, (50%) B is blue (56%). 50% * 56% = 28% chance of drawing this set.
3 - A is blue (50%), B is blue (44%). 50% * 44% = 22% chance of drawing this set.
4 - A is blue (50%), B is red (56%). 50% * 56% = 28% chance of draing this set.
Just to confirm that what we did is correct, remember that we must draw one of the four sets above (there aren't any other possibilities), meaning that there is a 100% of drawing one of those sets. So if we add the probabilities of drawing the sets together, they should equal 100. In fact, they do; 22 + 28 + 22 + 28 = 100.
Now going back to your original problem, with unordered sets. The only way it is different from the ordered sets above is that you can't tell the difference between set 2 (A=red, B=blue) and set 4 (A=blue, B=red), because you don't know which came first; you just see a red and a blue. With unordered sets then, the likelihood of getting one red and one blue is 56% (ie, the probability of set 2, above, plus the probability of set 4). The odds of getting red/red remain the same (22%), as do the odds of getting blue/blue (22%).
If my math is correct, you should be able to verify it experimentally. Put 5 red and 5 blue M&Ms in a bowl, and draw an unordered set (keep your eyes closed) of two M&Ms. Write down the colors you got. Put the two M&Ms back in the bowl, and repeat. Once you have repeated the steps above enough times, you should get 22% red/red, 22% blue/blue, and 56% being a combination of red and blue.
Or, if you want to experiment with ordered sets, just keep your eyes open, draw out your M&Ms one at a time, and write down which color come out first, and which second. The number of times you get each set should correspond to the probabilities listed for the four sets above.
Note that putting the M&Ms back in the bowl between drawings is very important. If you keep drawing once you've taken out the first two, you're not really repeating the experiment, you're just drawing out one really long set, and the colors you draw early in the set will effect which colors you draw later. The same principles discussed above will still apply, but you have to start computing the probabilities of ever larger sets. For example, if the first two you draw are red/red (remember there's a 22% chance of this) then only three of the eight M&Ms left in the bowl are red, so there is a 38% chance (actually 37.5%, but I'm rounding) of the third M&M being red. Thus the probability of drawing red/red/red equals 22% * 38%, which is 8%.
Who's the math nerd!? (Just kidding, because it's supposed to be me).
ReplyDeleteSince it's been so fully explained already, I won't go into the numbers. Although, I think the 2/3 probability for drawing two of the same color at the same time is correct because you only multiply probabilities in ordered sets and not when it is one event.
Anyhoo, it's been a while and I could be completely wrong.
The thing about probability, as you pointed out, is that it does seem to be based only on the observer, like the murderer example. And it can drive you nuts! I was good at it in the class or two I had, because some of it was obvious, but then there would be something that was completely unobvious and counter intuitive and how can that be!?!?!
I'll get my book out and let you have a look. Or maybe you need to do some meditation before you become a John Nash.
Brent, I do appreciate the thorough explanation. I really do. It just seems that probability doesn't mean anything in the "real" world. Or, rather, it doesn't really tell you anything more than you already know. In our M&M example, probability doesn't really seem to tell you anything other than: "There are 5 blue M&Ms and 5 red M&Ms." It just says it differently. But I guess that's what math is--identification.
ReplyDeleteIt's just funny to me that saying "a 50-50 chance" means nothing more than there are two possible outcomes, assuming something specific does not happen (which it must). That's the goofy yet ever so important rule behind it. We have to assume a state that probably (ha ha ha) doesn't exist. That is, that there is literally an equal chance among both/all options. There's not a 50-50 chance that I'll choose a red M&M no matter which M&M I choose. There's just a 50-50 chance I'll choose a red M&M up until I (i.e., for as long as I don't) make a choice (which, again, I must). At that point, there's just fact and fiction. So, in a very real sense, once someone has acted, probability is irrelevant.
(Remember, this is part of what baffles and aggravates me--because I realize probability does seem applicable and does seem worthwhile--but it simultaneously doesn't bear on anything in a literal way.)