Understanding ProbabilityDate: 05/13/2002 at 03:27:22 From: LuckyStar Subject: Mathematics - Probabilities I am doing a science project. If I were looking for a percentage or probability (e.g. for colored marbles) how would I go about it? To say it in a different way, say I were looking for a probability of colored marbles in three packages. How would I do that? Date: 05/13/2002 at 10:10:49 From: Doctor Ian Subject: Re: Mathematics - Probabilities Hi, The general idea is this. You start by defining a set of things that can happen, and identifying a subset of those things that are of interest to you. For example, suppose I flip a coin three times. If I use H to represent heads and T to represent tails, here are all the things that might happen: 1st 2nd 3rd toss toss toss H -> HH -> HHH HHT HT -> HTH HTT T -> TH -> THH THT TT -> TTH TTT In other words, there are 8 possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT Now, suppose I decide that I'm interested in tosses that result in exactly two heads. How many of those are there? HHH, HHT, HTH, HTT, THH, THT, TTH, TTT --- --- --- There are three. So the probability of getting exactly two heads is number of ways to get two heads p = ------------------------------------- number of ways to get anything at all 3 = --- 8 Now, what if I decide that I'm interested in exactly two of _either_ heads or tails. How many ways can that happen? HHH, HHT, HTH, HTT, THH, THT, TTH, TTT --- --- --- --- --- --- There are 6. So the probability is number of ways to get two heads or two tails p = -------------------------------------------- number of ways to get anything at all 6 = --- 8 So this is the basic idea behind probability. Where it gets tricky is this: As you start to involve more objects and more events, the numbers grow very quickly. For example, the number of things that can happen when you flip a coin N times is N Number of possibilities -- ----------------------- 1 2^1 = 2 2 2^2 = 4 3 2^3 = 8 4 2^4 = 16 5 2^5 = 32 6 2^6 = 64 10 2^10 = 1,024 20 2^20 = 1,048,576 30 2^30 = 1,073,741,824 And these are relatively small numbers, because a coin has only two sides it can land on. Suppose we roll a die N times. How many sequences can we get? N Number of possibilities -- ------------------------------------- 1 6^1 = 6 2 6^2 = 36 3 6^3 = 216 4 6^4 = 1,296 5 6^5 = 7,776 6 6^6 = 46,656 10 6^10 = 60,466,176 20 6^20 = 3,600,000,000,000,000 (approximately) 30 6^30 = 22,000,000,000,000,000,000,000 (approximately) Because the numbers grow so quickly, it becomes impossible to actually list all the possible events for all but the most trivial situations. Hence all the interest in coming up with formulas (like the ones you can find in our FAQ on "Permuations and Combinations") that can be used to compute probabilities directly! Also, probability often involves the use of tricks. One common trick is this: Sometimes it's easier to figure out how many ways something _can't_ happen than to figure out how many ways it _can_ happen. You can find an example of that here: http://mathforum.org/dr.math/problems/murad.04.03.02.html Because the numbers in probability grow so quickly, when you can't find a formula that matches your particular problem, often the best thing to do is play around with smaller versions of your problem (e.g., instead of using 20 marbles, use 2 or 3 or 4), looking for some kind of pattern that you can use to create a formula that you can use for the larger case. I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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