Two study questions

Danneedshelp · #1 $Old$ November 3rd, 2009, 10:12 PM

Q1: An urn contains 3 black, 4 red, and 5 white balls. Three balls are randomly drawn one at a time without replacement.
a. Find the probability all three balls are of the same color.
b. How many balls would you expect to be black or white?

A1: a) Let N=12, n=3, r=3(same color balls)

So, I used the above info to compute P(Y=3) using the hypergeometric dist and got 0.004545.

b) I am a little thrown off by the "or" in the question. Some help would be great.

Q2: Suppose Y , the number of fatal car accidents in a certain state, obeys a Poisson distribution with an average of four fatal accidents per day.
a. For a particular day, what is the probability that there is at least one fatal car accident?
b. Using Tchebysheff ’s thoerem, find an upper bound for P (Y
≥ 12).

A2: a) simple

b) I am stuck on this part. Here is some of work...

$\mu=\lambda=4$ and $\sigma=2$

Given $P(Y\geq\\12)\rightarrow$ by Tchebysheff's Theorem $P(|Y-4|)\geq\\k2)\leq\\\frac{1}{k^{2}}$ . So, k=4 and thus $P(|Y-4|)\geq\\8)\leq\\\frac{1}{16}$

That is what I have done so far, but I am not sure I am on the right track.

Any help would be great. I am trying to straighten some things out before my exam on Thursday.

Thanks

CaptainBlack · #2 $Old$ November 4th, 2009, 12:19 AM

Quote:

Originally Posted by Danneedshelp $View Post$

Q1: An urn contains 3 black, 4 red, and 5 white balls. Three balls are randomly drawn one at a time without replacement.
a. Find the probability all three balls are of the same color.
b. How many balls would you expect to be black or white?

A1: a) Let N=12, n=3, r=3(same color balls)

So, I used the above info to compute P(Y=3) using the hypergeometric dist and got 0.004545.

What is the probability of selecting three black balls? Three red balls? Three white balls?

CB

CaptainBlack · #3 $Old$ November 4th, 2009, 12:26 AM

Quote:

Originally Posted by Danneedshelp $View Post$

Q1: An urn contains 3 black, 4 red, and 5 white balls. Three balls are randomly drawn one at a time without replacement.
a. Find the probability all three balls are of the same color.
b. How many balls would you expect to be black or white?

A1: a) Let N=12, n=3, r=3(same color balls)

So, I used the above info to compute P(Y=3) using the hypergeometric dist and got 0.004545.

b) I am a little thrown off by the "or" in the question. Some help would be great.

There are now two types of ball 8 (black or white) balls and 4 red balls.

So the expected number of (black or white) balls is:

$E(n)=0 \times p(0)+1 \times p(1) + 2 \times p(2) + 3 \times p(3)$

where $p(r)$ is the probability of exactly $r$ (black or white) balls in three draws without replacement.

CB

Danneedshelp · #4 $Old$ November 4th, 2009, 02:47 PM

Quote:

Originally Posted by CaptainBlack $View Post$

What is the probability of selecting three black balls? Three red balls? Three white balls?

CB

So, If I let B represent the event of choosing three black balls, R the event of choosing three red balls, and W represent the event of choosing 3 white balls I need to find

$P(B\cup\\R\cup\\W)=P(B)+P(R)+P(W)$ by using the hypergeometric dist for each probability. So, I only need to change my r value each time. Correct?

Danneedshelp · #5 $Old$ November 4th, 2009, 02:54 PM

Quote:

Originally Posted by CaptainBlack $View Post$

There are now two types of ball 8 (black or white) balls and 4 red balls.

So the expected number of (black or white) balls is:

$E(n)=0 \times p(0)+1 \times p(1) + 2 \times p(2) + 3 \times p(3)$

where $p(r)$ is the probability of exactly $r$ (black or white) balls in three draws without replacement.

CB

The $r$ refers the collection of objects with the desired properties, so in this case, $r=8$ (black or white). Hence, each probability you showed above is hypergeomentric with $N=12, n=3$ , and $r=8$ ?

Thanks a lot for your help, I really appreciate it.

Can someone check my work for the Tchebysheffs propblem

CaptainBlack · #6 $Old$ November 4th, 2009, 03:13 PM

Quote:

Originally Posted by Danneedshelp $View Post$

So, If I let B represent the event of choosing three black balls, R the event of choosing three red balls, and W represent the event of choosing 3 white balls I need to find

$P(B\cup\\R\cup\\W)=P(B)+P(R)+P(W)$ by using the hypergeometric dist for each probability. So, I only need to change my r value each time. Correct?

You should not need to use the hypergeometric distribution explicitly, you should just be able to write these down.

Your P(B)=(3/12)(2/11)(1/10) etc.

CB

CaptainBlack · #7 $Old$ November 4th, 2009, 03:15 PM

Quote:

Originally Posted by Danneedshelp $View Post$

The $r$ refers the collection of objects with the desired properties, so in this case, $r=8$ (black or white). Hence, each probability you showed above is hypergeomentric with $N=12, n=3$ , and $r=8$ ?

Thanks a lot for your help, I really appreciate it.

Can someone check my work for the Tchebysheffs propblem

No that is not what I wrote, and I did mean exactly what I did write.

CB