Q. 6.4 - A First Course in Probability

Question

Repeat Problem $6.2$ when the ball selected is replaced in the urn before the next selection.

Step-by-Step Solution

Answer

The table is,

Step 1: Introduction

A joint probability is a statistical measure that determines the chance of two events occurring at the same time and in the same place. 

The possibility of an event $Y$ occurring at the same time as an event $X$is known as joint probability.

Step 2 : Explanation of part(a)

The selected ball is replaced in the urn before the next selection.

Let$X_i$ equal $1$ if the ith white ball is selected $0$ otherwise.

Let the first and second ball chosen be white.

Such that

$$\begin{gathered} X_1=1, \\ {X}_2=2 \end{gathered}$$

If the first ball chosen is white,

The probability of the event is $5 / 13.$

The selected white ball is replaced in the urn.

The second chosen ball is also white and due to replacement, the probability remains the same.

Now,

Table representing joint probability for $X_1$ and $X_2$ :

Simplify the above table:

Step 3: Explanation of part(b)

Before the next selection, the picked ball is replaced in the urn.

If the ith white ball is selected, set $X i$to $1$; otherwise, set it to $0$.

All conceivable scenarios must be investigated and the combinatory argument must be used, using the same notion as in Part (a).

Table with joint probabilities for $X 1$, $X 2$, and $X 3$:

Simplify the above table: