Suppose that we throw balls into m bins until some bin contains
two balls. Each throw...
Suppose that we throw balls into m bins until some bin contains
two balls. Each throw
is independent, and each ball is equally likely to end up in any
bin. What is the expected number of
ball throwed?
Suppose that we have two bags each containing black and white
balls. One bag contains two times as many white balls as black
balls. The other bag contains two times as many black balls as
white. Suppose we choose one of these bags at random. For this bag
we select five balls at random, replacing each ball after it has
been selected. The result is that we find all balls are white
balls. What is the probability that we were...
13. A bin contains 3 red and 4 green balls. 3 balls are chosen
at random, with replacement. Let the random variable X be the
number of green balls chosen. a. Explain why X is a binomial random
variable. b. Construct a probability distribution table for X. c.
Find the mean (expected value) of X. d. Use the law of Large
Numbers to interpret the meaning of the expected value of X in the
context of this problem.
Suppose we have seven identical balls to be distributed in bins
labeled A, B, C, and D. For example, one way to
distribute the balls is to place two in A, none in B, four in C,
and one in D.
a) How many ways are there to distribute the balls among the
four bins? Explain your answer.
b) How many ways are there to distribute the balls so that at
each bin has at least one ball in it? Explain...
We have a bag that contains n red balls and n blue balls. At
each of 2n rounds we remove one of the balls from the bag randomly,
and place it in one of available n bins. At each round, each one of
the balls that remain in the bag is equally likely to be picked, as
is each of the bins, independent of the results of previous rounds.
Let Nk be the number of balls in the k-th bin...
An urn contains n white balls and m black balls. ( m and n are
both positive numbers.)
(a) If two balls are drawn without replacement , what is the
probability that both balls are the same color?
(b) If two balls are drawn with replacement (i.e., One ball is
drawn and it’s color recorded and then put back. Then the second
ball is drawn.) What is the probability that both balls are the
same color.
(c) Show that the...
An urn contains colored balls;5 red balls, 8 green balls, and 10
blue balls. Suppose If the 3 balls are drawn one after another
without replacement, what is the probability that the colors
observed will be Red, Green, Blue in this order? If the three
balls are drawn simultaneously from the urn (without replacement),
what is the probability that the selected balls will be all
different?
Suppose that there are two hats in front of you.
Hat 1 contains 4 green balls and 6 orange balls.
Hat 2 contains 3 green balls, 6 orange balls, and a purple
ball.
(a) Suppose you draw one ball from each hat. The outcome of
interest is the colour of each of the drawn balls. How many
elements are in the sample space of this experiment?
(b) Write out the complete sample space for the experiment
above.
(c) What is...
A coin has a probability of 1/4 for head, and is repeatedly
tossed until
we throw head. The successive results of the toss are independent
of each other.
What is the probability that the first time we throw head after an
odd number of toss?
Hint: Use the law of total probability and consider the event that
the
first toss is head is, and her complement, as conditioning
events.
Correct answer: 3/7
Bin packing, or some variation of bin packing, can be found in
different applications. Use two examples, berth allocation and VRP,
to discuss why in some scenarios, these two problems are actually
equivalent to bin packing.
A bucket contains three red balls and four white balls. Two
balls are drawn from the bucket in succession without replacement.
What is the probability of getting two white balls, ie. a white
ball on the first draw followed by a white ball on the second
draw?