Thread: Explain This Probability Word Problem to me (Normal Distribution)

1. Explain This Probability Word Problem to me (Normal Distribution)

Suppose that the wait time in the local coffe shop ranges evenly between 0 and 300 seconds.

a.) what is the probability that you will wait less than 100 seconds to be served?

b.) What is the probability that you will wait more than two minutes to be served?

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The wait time is a uniform random variable X between 0 and 300s. The probability density function is

f(x) = 1 / (300-0) = 1/300

Probability that wait time is less than 100s is

P(X<100s) = (x:0 to 100) ∫f(x)dx = (x:0 to 100) ∫(1/300)dx = [x/300](x:0 to 100) = 100/300 - 0/300 = 1/3

Probability that wait time is more than 2min (= 120s) is

P(X>120s) = (x:120 to) ∫f(x)dx = (x:120 to 300) ∫(1/300)dx = [x/300]{x:120 to 300} = 300/300 - 120/300 = 180/300 = 0.6
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I don't just want to "Write down" the "answers". Can someone please explain/show me what is going on here?
"p(x<100)" I understand, where the probability of x is less than 100. I don't understand what is happening after that, in both situations. What is a density function? I'm confused.

2. Re: Explain This Probability Word Problem to me (Normal Distribution)

If, even after someone "gave" you that solution, you still don't understand it, where did you get this problem? It looks like a typical homework problem in basic probability but you seem to be saying that you have never taken such a course. Do you know what "ranges evenly" or "uniform distribution" mean? It is particularly strange that you titled this "Normal Distribution" but then use a uniform distribution.

3. Re: Explain This Probability Word Problem to me (Normal Distribution)

Originally Posted by HallsofIvy
If, even after someone "gave" you that solution, you still don't understand it, where did you get this problem? It looks like a typical homework problem in basic probability but you seem to be saying that you have never taken such a course. Do you know what "ranges evenly" or "uniform distribution" mean? It is particularly strange that you titled this "Normal Distribution" but then use a uniform distribution.
Well, I'm actually doing this course in a book. (I get the luxury of teaching myself .)This is one of the questions and it was in the section labelled "Probability Distribution". Uniform distribution would be where all the outcomes have a uniform (in other words equal) chance of being chosen. Ranges evenly.... uhmm... not so sure, does it not mean exactly what it says "ranging" "evenly"? Maybe I was wrong in my title, labelling it "Normal Distribution" as that would be something that is symmetrical on the mean? I know this solution was given to me, and I agree that I don't understand it, which is why I'm asking for help! I think you've answered questions for me before, your name seems familiar lol.

4. Re: Explain This Probability Word Problem to me (Normal Distribution)

Originally Posted by tdotodot
Maybe I was wrong in my title, labelling it "Normal Distribution" as that would be something that is symmetrical on the mean? I know this solution was given to me, and I agree that I don't understand it, which is why I'm asking for help! I think you've answered questions for me before, your name seems familiar lol.
Yes, that title is misleading. This is uniform distribution.

We say that $\displaystyle X$ is uniformly distributed on $\displaystyle [a,b]$ if its pdf is:

$\displaystyle f(x) = \left\{ {\begin{array}{rl} {\dfrac{{1}}{{b - a}},}&{x \in \left[ {a,b} \right]} \\ {0,}&{\text{else}} \end{array}} \right.$

So that is what you have here.

5. Re: Explain This Probability Word Problem to me (Normal Distribution)

Originally Posted by Plato
Yes, that title is misleading. This is uniform distribution.

We say that $\displaystyle X$ is uniformly distributed on $\displaystyle [a,b]$ if its pdf is:

$\displaystyle f(x) = \left\{ {\begin{array}{rl} {\dfrac{{1}}{{b - a}},}&{x \in \left[ {a,b} \right]} \\ {0,}&{\text{else}} \end{array}} \right.$

So that is what you have here.
Uniform Distribution was in my previous unit, however it did not involve much formulas. In this unit the book mentioned a formula for Normal Distribution, For a normal curve:

y= 1/ (standard deviation) √ 2pi *e ^-1/2*x-mean/standard deviation^2 ... but it says it is "beyond the realm of of his course".

I think the equation you provided although relevant is beyond what I am currently learning, but I may be wrong? Is there a more simple or different way to solve this?

6. Re: Explain This Probability Word Problem to me (Normal Distribution)

Originally Posted by tdotodot
Uniform Distribution was in my previous unit, however it did not involve much formulas.
That may have been a previous unit. But what posted is a uniform distribution problem.

Originally Posted by tdotodot
Suppose that the wait time in the local coffe shop ranges evenly between 0 and 300 seconds.
a.) what is the probability that you will wait less than 100 seconds to be served?
b.) What is the probability that you will wait more than two minutes to be served?
a) $\displaystyle \frac{100-0}{300-0}$
b) $\displaystyle \frac{300-120}{300-0}$