question about probability
If an urn contains 100 green or red marbles, and you sample 10 and they are all green, what is the probability that they are all green?

Re: question about probability
Quote:
Put it to the simplest case, if you flip a coin and get a head, what is the head probability of the coin? The answer can be "any nonzero number" but there is no further information to pin it down. Hope this helps. 
Re: question about probability
Quote:
Do these help? Read through the second one for a similar example. http://badmomgoodmom.blogspot.com/20...rtone_16.html http://badmomgoodmom.blogspot.com/20...parttwo.html 
Re: question about probability
Does the Hoeffding Inequality allow us to say something about this probability?
P[Ein  Eout > epsilon] <= 2e^(2 * epslion^2 * N) Since Ein = 0, N = 10, setting epsilon to 0.5, the inequality gives us: P[Eout > 0.5] <= 2e^(5) = 0.013+ This seems to be saying something nontrivial about Eout. 
Re: question about probability
I would think this way:
if p is probability of the outcome then for 10 trials there is p^10 probability of getting all favorable outcome. 
Re: question about probability
Hoeffding's inequality has a free parameter in your question, namely . It lest you say that since you have 10 samples, if you want to be 80% sure of the distribution of marbles in the jar (), then the jar is at most percent different from the sample.
where substituting from your example: simplifying: so we think that with 80% confidence, the jar is at least 10.2% green. We probably need a bigger N! 
Re: question about probability
In the second lecture, the Professor asked a question about flipping a coin:
What is the probability of getting all ten heads if you flip a coin 10 times count the number of heads, and then you repeat the experiment 1000 times. The answer he gave was 63%  I would like to know how this was computed. Any help would be greatly appreciated. Thanks! 
Re: question about probability
Quote:
Therefore, the probability of not getting 10 heads for one coin is approximately . This means that the probability of not getting 10 heads for any of 1000 coins is this number multiplied by itself 1000 times, once for every coin. This probability is therefore . This is approximately since . Numerically, . Therefore, the probability of this not happening, namely that at least one coin of the 1000 coins will give 10 heads, is 1 minus that. This gives us the answer of approximately 0.63 or 63% that I mentioned in the lecture. 
Re: question about probability
Thank you Professor, that was extremely helpful.

Re: question about probability

Re: question about probability
The law of big numbers states that the average $\nu_min$ is close to the $E{\nu_min}$.
$E\nu_min$ can be calculated directly for this experiment. $P(\nu_min=0)$=0.623576 $P(\nu_min = 0.1)$ = 0.3764034 $P(\nu_min = 0.2)$ = 0.00002; and $P(\nu_min>=0.3)=0$ for the purposes of calculating the mean. Therefore, $E(\nu_min)$=0.037644, and the average proportion of heads for c_min should be close to this number. 
Re: question about probability

Re: question about probability
Quote:

Re: question about probability
Quote:

Re: question about probability
how did you calculate those probability values? ( nu_min = 0, 0.1, 0.2 )

Re: question about probability
Thank you, Professor.
This how I calculate the probabilities. Let  the number of heads for , and let be the number of heads in ith experiment (out of 1000). Then, as Professor has shown previously, Now, . Therefore, Next, Next, The rest can be calculated directly too, but they are essenctially 0 for the purpose of calculating the mean. 
Re: question about probability
thank you for the detailed explanation.

Re: question about probability
the answer is very clear but how do we know when to use this not.
to clarify my question, if say P(ten heads)=p and P(not ten heads)=q (=1p). why does using (p^1000) give the wrong answer? 
Re: question about probability
Quote:

Re: question about probability
Quote:

Re: question about probability
Quote:
Probability of all heads in 10 flips by one coin: p = 0.5^10 Probability of all heads in 10 flips by any of 1000 coin: 1000*p = 97.7% In above I assumed there were 1000 10flips. 
Re: question about probability
Quote:

Re: question about probability
Thank you very much, Professor. The explanation is spot on, very helpful.

All times are GMT 7. The time now is 07:04 PM. 
Powered by vBulletin® Version 3.8.3
Copyright ©2000  2022, Jelsoft Enterprises Ltd.
The contents of this forum are to be used ONLY by readers of the Learning From Data book by Yaser S. AbuMostafa, Malik MagdonIsmail, and HsuanTien Lin, and participants in the Learning From Data MOOC by Yaser S. AbuMostafa. No part of these contents is to be communicated or made accessible to ANY other person or entity.