ATAR Notes: Forum

VCE Stuff => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematics => Topic started by: /0 on September 03, 2009, 09:46:53 pm

Title: probability questions
Post by: /0 on September 03, 2009, 09:46:53 pm: "Given the number of days in a month is 30, what is the probability that the number of letters in the name is less than or equal to 6?"

Do they seem like dependent or independent events? How can you tell quickly?
Title: Re: probability questions
Post by: Mao on September 03, 2009, 10:36:53 pm: April, June, September, November
Pr = 0.5
...? [I don't see any other way of doing it..]
Title: Re: probability questions
Post by: /0 on September 03, 2009, 10:51:25 pm: Hmm it's just that the book I'm using says that they are independent events, but intuitively they seem to be dependent... i.e. the information in the first clause 'seems' to help you identify a more accurate probability.

Apart from evaluating $Pr(a \cap b) = Pr(a)Pr(b)$ I wouldn't have known they were independent events, would you?
Title: Re: probability questions
Post by: /0 on September 03, 2009, 11:15:29 pm: Also, I'm having trouble with this problem:

"The future lifetimes (in months) of two components of a machine have the following joint density function.

$f_{XY}(x,y) = \begin{cases} \frac{6}{125000}(50-x-y) \ \ , \ \ 0<x<50-y<50 \\\\ 0 \ \ , \ \ \mbox{otherwise}\end{cases}$

What is the probability that both components are still functioning 20 months from now?"

The answer is 0.008 but I get 0.424 :O

Is my area interpretation correct?

thanks
Title: Re: probability questions
Post by: Mao on September 04, 2009, 09:25:55 am: White: 0< x < 20, 0 < y < 20
Green: 0< x < 20, 20 < y < 50
Blue: 20 < x < 50, 0 < y < 20
Red: 20 < x, 20 < y

You can see that green and blue does not meet the conditions of x > 20, y > 20, hence only red is the region required
Title: Re: probability questions
Post by: /0 on September 04, 2009, 12:00:19 pm: thanks maaaoooo!
Title: Re: probability questions
Post by: /0 on September 05, 2009, 04:57:09 am: There's a proof in the book that $Pr(X \geq \mu+a )\leq \frac{\sigma^2}{\sigma^2+a^2}$ and it involves the step:

$\frac{E[(X+b)^2 ]}{(a+b)^2}=\frac{\sigma^2+b^2}{(a+b)^2}$

I don't get how that works though.... how does it simplify?

Here's the proof: