Question

In: Statistics and Probability

If each of n balls is placed at random in k urns, what is the probability...

If each of n balls is placed at random in k urns, what is the probability that exactly two urns remain empty?

Solutions

Expert Solution

Provided k<n

Total no. of ways each of n balls can be placed at random in k urns = kn

Now consider that only the first two cells remain empty.

This can be achieved by first putting one ball in each of the remaining k-2 urns, and then placing the remaining n-(k-2)=n-k+2 balls in the k-2 urns in (k-2)n-k+2 ways.

This event can happen for any two urns, which can be selected in ways.

Total no. of ways each of n balls can be placed at random in k urns in such a way that exactly 2 urns remain empty


Related Solutions

If 12 balls are thrown at random into 20 boxes, what is the probability that no...
If 12 balls are thrown at random into 20 boxes, what is the probability that no box will receive more than one ball? Please explain.
There are 7 balls numbered 1 through 7 placed in a bucket. What is the probability...
There are 7 balls numbered 1 through 7 placed in a bucket. What is the probability of reaching into the bucket and randomly drawing two balls numbered 6 and 3 without replacement, in that order? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
There are two urns that, between them, contain five balls. At each time step, one of...
There are two urns that, between them, contain five balls. At each time step, one of the five balls is moved to the other urn. Let the state variable be the number of balls in Urn 1. Find the fixed vector. a) Draw a state transition diagram and find the transition matrix. b) Is this a regular chain? Is this an ergodic chain? c) Find the fixed vector. What is the probability that in the long run Urn 1 has...
Fifteen DVDs are placed on a shelf in random order. a) What is the probability that...
Fifteen DVDs are placed on a shelf in random order. a) What is the probability that you favourite DVD is first or your least favourite DVD is last? b) How would the probability change if you would not want both your favourite DVD to be first and your least favourite DVD last?
There are three urns each containing seven red, five green, and three white balls, and two...
There are three urns each containing seven red, five green, and three white balls, and two old urns each containing five red, three green, and seven white balls. The urns are identical except for an old or new date stamped beneath the base. If a single red ball is randomly drawn from one of these urns, was it most probably drawn from an old urn or a new urn?
What is the number of ways to distribute 12 identical balls in 8 different urns, so...
What is the number of ways to distribute 12 identical balls in 8 different urns, so that two of the urns will contain together at least 10 balls?
Distribute 13 indistinguishable balls in 6 distinguishable urns. What is the number of distributions in which...
Distribute 13 indistinguishable balls in 6 distinguishable urns. What is the number of distributions in which the first three cells contain together AT LEAST 10 balls? What would be the answer if the balls were distinguishable?
We have a bag that contains n red balls and n blue balls. At each of...
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...
Consider two urns of balls: the first contains 5 different red balls numbered from 1 to...
Consider two urns of balls: the first contains 5 different red balls numbered from 1 to 5 and the second contains 4 different blue balls numbered from 1 to 4. You are asked to pick one ball from the first urn (i.e., the one with red balls) and one ball from the second urn (i.e., the one with blue balls). Each outcome has the form (r, b), where r denotes the number on the red ball and b denotes the...
Three white and three black balls are distributed in two urns in such a way that...
Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state i,i = 0, 1, 2, 3, if the first urn contains i white balls. At each step, we draw one ball from each ufn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. Let Xq denote the state of the...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT