1. ## finding the probability

A bombing plane flies directly above a railroad track. Assume that if a large (small) bomb falls within 40 (15) feet of the track, the track will be sufficiently damaged so that the traffic will be disrupted. Let X denote the perpendicular distance from the track that a bomb falls. Assume that
[IMG]file:///C:/Users/Standard/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG] fx(x) = (100 - x)/5000 I(0, 100) (x)

There are two problems. I would like somebody to check my work for this first problem.

(a) Find the probability that a large bomb will disrupt traffic.
[IMG]file:///C:/Users/Standard/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
I found the cdf of X by integrating the f(t) dt from -infinity to x
Actually from 0 to x integration of (100-t)/5000. Then I got the cdf as F(x) = 1/5000 [100x - (x^2/2)]

I used the cdf to find the probability between 0 and 40.
P(0 < x< 40) = P(x<40) - P(x<0) = F(40) - F(0)
Finally, I got the answer as 0.64
I am wondering why I have not used the interval 0<x<100.

I don't have an idea on how to solve the second problem that follows.
(b) If the plane can carry three large (eight small) bombs and uses all three (eight), what is the probability that the traffic will be disrupted?

2. Originally Posted by cielo
A bombing plane flies directly above a railroad track. Assume that if a large (small) bomb falls within 40 (15) feet of the track, the track will be sufficiently damaged so that the traffic will be disrupted. Let X denote the perpendicular distance from the track that a bomb falls. Assume that
[IMG]file:///C:/Users/Standard/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG] fx(x) = (100 - x)/5000 I(0, 100) (x)

There are two problems. I would like somebody to check my work for this first problem.

(a) Find the probability that a large bomb will disrupt traffic.
[IMG]file:///C:/Users/Standard/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
I found the cdf of X by integrating the f(t) dt from -infinity to x
Actually from 0 to x integration of (100-t)/5000. Then I got the cdf as F(x) = 1/5000 [100x - (x^2/2)]

I used the cdf to find the probability between 0 and 40.
P(0 < x< 40) = P(x<40) - P(x<0) = F(40) - F(0)
Finally, I got the answer as 0.64
I am wondering why I have not used the interval 0<x<100.

I don't have an idea on how to solve the second problem that follows.
(b) If the plane can carry three large (eight small) bombs and uses all three (eight), what is the probability that the traffic will be disrupted?

You cannot insrt image files from your c: drive, they are not accessible across the internet, they must be uploaded to a location from which they can be linked to.

CB

3. Does it mean, anyone can't read my posted problem?

4. I am wondering why I have not used the interval 0<x<100.
You are interested in the probabiliity of the bomb falling within 40 feet of its target. I am not sure where you got the number 100 from.

Part b
for part (b) you can use the binomial distribution. You will have to assume that the outcome of each bomb drop is independant.

For large bombs:
You have 3 trials, and you worked out the probability of success on each trial in part (a)

For small bombs:
You have eight trials, and you need to work out the probability of success on each trial. Use the same method as part (a)

5. Originally Posted by cielo
A bombing plane flies directly above a railroad track. Assume that if a large (small) bomb falls within 40 (15) feet of the track, the track will be sufficiently damaged so that the traffic will be disrupted. Let X denote the perpendicular distance from the track that a bomb falls. Assume that
[IMG]file:///C:/Users/Standard/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG] fx(x) = (100 - x)/5000 I(0, 100) (x)

There are two problems. I would like somebody to check my work for this first problem.

(a) Find the probability that a large bomb will disrupt traffic.
[IMG]file:///C:/Users/Standard/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
I found the cdf of X by integrating the f(t) dt from -infinity to x
Actually from 0 to x integration of (100-t)/5000. Then I got the cdf as F(x) = 1/5000 [100x - (x^2/2)]

I used the cdf to find the probability between 0 and 40.
P(0 < x< 40) = P(x<40) - P(x<0) = F(40) - F(0)
Finally, I got the answer as 0.64
I am wondering why I have not used the interval 0<x<100.

I don't have an idea on how to solve the second problem that follows.
(b) If the plane can carry three large (eight small) bombs and uses all three (eight), what is the probability that the traffic will be disrupted?

The pdf of the perpendicular distance of the bomb impact from the road is:

$f_x(x)=\begin{cases} \dfrac{100-x}{5000} &,\ x \in (0,100)\\ 0 & ,\ \text{ otherwise} \end{cases}$

You should check that this is a pdf by integrating it form $x=0$ to $x=100.$

Now the probabilty of disrupting the traffic is:

$\displaystyle p=\int_0^{40} f_x(x)\; dx$

which you should evaluate.

CB

6. Originally Posted by CaptainBlack
The pdf of the perpendicular distance of the bomb impact from the road is:

$f_x(x)=\begin{cases} \dfrac{100-x}{5000} &,\ x \in (0,100)\\ 0 & ,\ \text{ otherwise} \end{cases}$

You should check that this is a pdf by integrating it form $x=0$ to $x=100.$

Now the probabilty of disrupting the traffic is:

$\displaystyle p=\int_0^{40} f_x(x)\; dx$

which you should evaluate.

CB
Thank you for following up my posted problem! I am also happy for posting the pdf in a very good structure. I was trying to do that one first in a word document, but they changed in form when I copied the equations here.