Probability related problems

Mechanics, discrete, statistics, ...
PurpleBlu3s
Posts: 73
Joined: Mon Sep 19, 2011 5:49 pm

Probability related problems

I'm rather stumped with the PE problems about probability where it says something like "give your answer to 8 decimal places", and is looking for a quantity that can only be an integer in any one instance. I just have no idea what to look at - in all my attempts and with all my (limited) knowledge of probability I can only ever find integer solutions. I have tried looking up probability but I haven't found anything that helps with the PE style questions.

I would appreciate some kind of subtle pointer to an area I could research if possible (maybe I'm just missing something obvious, but general probability from school doesn't allow you to solve those problems).

Many thanks. jaap
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Re: Probability related problems

I think the problem you are having is not realising that Expected Value has a very specific meaning. The explanation on wiki is pretty good:
... the expected value ... of a random variable is the weighted average of all possible values that this random variable can take on. The weights used in computing this average correspond to the probabilities in case of a discrete random variable ...
More informally, it can be interpreted as the long-run average of the results of many independent repetitions of an experiment (e.g. a dice roll). The value may not be expected in the ordinary sense—the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), just like the sample mean.
The expected value of a single dice roll would be 3.5, since that is (1+2+3+4+5+6)/6 even though you obviously cannot throw a value of 3.5.
Jaap's Puzzle Page PurpleBlu3s
Posts: 73
Joined: Mon Sep 19, 2011 5:49 pm

Re: Probability related problems

Thanks. I did not realise what expected value actually meant (especially when written "expected number of ...").

Making it work with conditional probabilities seems tricky though - for the problem I'm trying, I'm getting rather nonsensical answers. :< dharasty
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Joined: Wed Apr 25, 2012 2:32 pm

Re: Probability related problems

Perhaps to some, the term "expected value" MIGHT sound like some one is asking: what is the "most likely value". But these are not the necessarily same.

If you roll two six-sided dice, the mostly likely value is 7... and the expected value is also 7. However, if you roll a single six-sided die, all possible outcomes are equally likely... but the expected value (as described above) is 3.5.

FYI: here's another way to think about "expected value" that should be understandable by just about anyone (no formal probability training required):
• If you perform the stated activity that has a numerical outcome only once (such as some experiment, or dice roll, or watch a random event), then you get one of the possible outcome values.
• We could also ask you to do the activity 100 times, and ask you to average your values. If you rolled a six sided die 100 times and totaled the outcomes, you may get 372 as the answer, or an average of 3.72 per die roll.
• Do the same for 10000 rolls, and the average may be 3.4812.
• Loosely: The "expected value" is the "average" if you performed the activity an "infinite" number of times.
Which of course, you can't do. But there are other ways to calculate the expected value.
• In principle all you have to do is find the SUM of: each possible value times the probability of that value.
• For rolling a six sided die, this only takes a few arithmetic steps.
• Sometimes, even for activities which might have an infinite number of steps, a little mathematical handiwork can get a single answer. (Example: "Flip a coin. If you get heads, continue. If you get tails, stop. What is the expected number of heads?" ANY NUMBER is possible. So you have to sum up an infinite series of numbers. But still, this has a finite expected value.)
• For ProjectEuler problems... one often doesn't have the luxury of immediately knowing the probabilities... or even a clear idea of the valid outcomes... or the possible outcomes may be some real number (rather than integer)... ah, but that's where the fun starts. 