Chapter 4: Q.4.10 (page 173)

An urn contains $n$ balls numbered $1$ through $n$ . If you withdraw $m$ balls randomly in sequence, each time replacing the ball selected previously, find $P {X = k}, k =$ $1, . . ., m$
where $X$ is the maximum of the $m$ chosen numbers.
Hint: First find $P {X \leq k}$ .

Short Answer

Expert verified

$P (X \leq k) = {(\frac{k}{n})}^{m}$

$P (X = k) = {(\frac{k}{n})}^{m} - {(\frac{k - 1}{n})}^{m}$

Step by step solution

Step 1: Given information

An urn contains n balls numbered $1$ through $n$ . If you withdraw $m$ balls randomly in sequence, each time replacing the ball selected previously.

Step 2: Explanation

For the beginning, we will find $P (X \geq k)$ . Observe that in total, there exist $n^{m}$ sequences of drawn balls. If $X \leq k$ , that means all balls in the sequence that has been drawn must be less or equal to $k$ . There exist $k^{m}$ of these sequences. Hence

$P (X \leq k) = \frac{k^{m}}{n^{m}} = {(\frac{k}{n})}^{m}$

Now, we have that

$P (X = k) = P (X \leq k) - P (X \leq k - 1) = {(\frac{k}{n})}^{m} - {(\frac{k - 1}{n})}^{m}$

Step 3:Final answer

$P (X \leq k) =$ ${(\frac{k}{n})}^{m}$

$P (X = k)$ $= {(\frac{k}{n})}^{m} - {(\frac{k - 1}{n})}^{m}$

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An urn contains $n$ balls numbered $1$ through $n$ . If you withdraw $m$ balls randomly in sequence, each time replacing the ball selected previously, find $P {X = k}, k =$ $1, . . ., m$
where $X$ is the maximum of the $m$ chosen numbers.
Hint: First find $P {X \leq k}$ .

Short Answer

Step by step solution

Step 1: Given information

Step 2: Explanation

Step 3:Final answer

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An urn containsnballs numbered 1through n. If you withdraw mballs randomly in sequence, each time replacing the ball selected previously, findP{X=k},k=1,...,mwhere Xis the maximum of the mchosen numbers.Hint: First find P{X≤k}.

Short Answer

Step by step solution

Step 1: Given information

Step 2: Explanation

Step 3:Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Pure Maths

Calculus

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

An urn contains $n$ balls numbered $1$ through $n$ . If you withdraw $m$ balls randomly in sequence, each time replacing the ball selected previously, find $P {X = k}, k =$ $1, . . ., m$
where $X$ is the maximum of the $m$ chosen numbers.
Hint: First find $P {X \leq k}$ .