Q. 2.1 - A First Course in Probability

Question

A box contains $3$ marbles: $1$ red, $1$ green, and $1$ blue. Consider an experiment that consists of taking $1$ marble from the box and then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without replacing the first marble.

Step-by-Step Solution

Answer

The sample space of drawing one marble with replacement is $\left\{(R,R),(R,G),(R,B),(G,R),(G,G),(G,B),(B,R),(B,G),(B,B)\right\}$

The sample space of drawing one marble without replacement is $\left\{(R,G),(R,B),(G,R),(G,B),(B,R),(B,G)\right\}$

Step 1. Given information.

It is given that,

Total no. of marbles $=3$

No. of Red Marbles$=1$

No. of Green Marbles$=1$

No. of Red Marbles$=1$

Let $R$ represent the red marble, $G$represent the green marble and $B$ represent the blue marble.

Step 2. Find the sample space of drawing one marble with replacement.

The sample space consists of all possible results of this experiment.

Let us assume that the red ball is drawn first, the second ball can de either red, or blue, or green and so on.

The size of sample space will be $3\times 3$. Since, there are three possibilities for the first draw and three possibilities for the second draw.

Thus, the sample space is:

$\left\{(R,R),(R,G),(R,B),(G,R),(G,G),(G,B),(B,R),(B,G),(B,B)\right\}$

Step 3. Find the sample space of drawing one marble without replacement.

The sample space consists of all possible results of this experiment.

Let us assume that the red ball is drawn first, the second ball can de either blue, or green and so on.

The size of sample space will be $3\times 2$. Since, there are three possibilities for the first draw and two possibilities for the second draw.

Thus, the sample space is:

$\left\{(R,G),(R,B),(G,R),(G,B),(B,R),(B,G)\right\}$