Question

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

Short answer

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 event Y occurring at the same time as event X is known as joint probability.

Step-by-step solution

Introduction

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 event Y occurring at the same time as event X is known as joint probability.

Explanation of (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

localid="1647491617162" $$\begin{gathered} X_1=1, \\ {X}_2=2 \end{gathered}$$

If the first ball chosen is white,

The probability of the event islocalid="1647491628607" $5/13.$

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 localid="1647491637693" $X_1$and localid="1647491645104" $X_2$:

Simplify the above table:

Explanation of (b)

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

If the ith white ball is selected, set $Xi$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 $X1$, $X2$, and $X3$:

Simplify the above table: