IE 111 Fall 2008
Homework #6
Due Wednesday October 22
Question 1.
To check on the security screeners at an airport, we put 4 fake bombs into a set of 52
baggage pieces.
Suppose that the screeners actually just pick 6 pieces randomly (without
replacement) to inspect.
Let X be the number of our fake bombs that they find.
a)
According to which well known distribution is X distributed?
b)
Find the Probability Mass Function f(x).
Remember to specify an appropriate range for x.
c)
Find E(X)
d)
Find V(X)
Question 2.
An urn has 6 red marbles and 14 green marbles.
Careful on this one, the distribution to
use keeps changing!
a)
If I pick 3 marbles
with replacement
, and let X= the number of red marbles picked,
what is the P(X=1)?
b)
If I keep picking marbles
with replacement
until I get my first red marble, what is the
probability I picked exactly 7 times?
c)
If I pick 3 marbles
without replacement
and let X= the number of red marbles picked,
what is the P(X=1)?
d)
If I pick 3 marbles
without replacement
and let X= the number of red marbles picked,
what is the expected value of X?
e)
Suppose I have picked 8 marbles
with replacement
and have not yet gotten a red
marble.
How many more picks do I expect will be required before my first red
marble?
f)
Suppose I have picked 9 marbles
without replacement
and have not yet gotten a red
marble.
How many more picks (without replacement) do I expect will be required
before my first red marble?
This is a tricky problem!
We can't use any of our favorite
distributions; you have to think through it from first principles.
Here's a hint to get
you started.
Let X = number of remaining picks.
Compute P(X=1), P(X=2), P(X=3),
etc. (it will help to think of the Geometric distribution, but it's not quite the same) and
then compute the mean value from there.
Question 3.
Suppose you go shopping at a convenience store, pick your items, and then join the line for the cash
register. There are 6 people in the line ahead of you, plus one person who is currently being served (there
is only one clerk on duty).
Having nothing better to do, you estimate that the time it takes to ring up one
person’s items (measured in seconds) has a Geometric distribution with parameter p=1/30.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Storer
 Probability theory, red marble, cent drinks

Click to edit the document details