A jar contains $36$disks: $9$each of four colors—red, green, blue, and yellow. Each set of disks of the same color is numbered from $1to9$. Suppose you draw one disk at random from the jar. Define events R: get a red disk, and N: get a disk with the number $9$.

a. Make a two-way table that describes the sample space in terms of events R and N.

b. Find P(R) and P(N).

c. Describe the event “R and N” in words. Then find the probability of this event.

d. Explain why$P(RorN)\ne P\left(R\right)+P\left(N\right)$Then use the general addition rule to compute P(R or N).

R: get a red disk

N: get a disk with the number$9$

For two – way table,

We are required to make two rows and two columns.

Let us name the rows and columns according to the events R and N.

Rows: red, not red.

Columns: nine, not nine.

Since there are $36$disks in total, we have to mention 36 in the bottom left corner of the table.

We know that

There are $9$disks of red, blue, green and yellow each.

This implies

There are $9$disks which are red in color.

And

Remaining $27$disks of blue, green and yellow color.

We also know that

Each color disks are numbered $1-9$.

This implies

There is only $1$disk which is colored red and numbered nine.

And

Remaining $3$disks are numbered $9$and of different color.

This also implies

With only $1$disk of number $9$and colored red, remaining $8$disks of red color are numbered $1-8$.

Since $3$disks of different color are numbered $9$, remaining $24$disks of different color are numbered $1-8$.

Two – way table is shown as:

R: get a red disk

N: get a disk with the number$9$

We know that

$9$out of $36$disks are red in color.

When number of favorable outcomes are divided by the number of possible outcomes, we get the probability.

role="math" localid="1663581576492" $P\left(R\right)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes}=\frac{9}{36}=\frac{1}{4}=0.25$

We also know that

Each color disks are numbered $1-9$.

With $4$colors, $4$disks are numbered $9$among $36$disks.

$P\left(N\right)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes}=\frac{4}{36}=\frac{1}{9}\approx 0.1111$

R: get a red disk

N: get a disk with the number$9$

“R and N” represents the event that the disk is numbered $9$and red in color.

We know that

Out of total $36$disks, there are $9$disks which are red in color.

Since each color disks are numbered $1-9$.

This implies

There is only one disk which is numbered $9$and red in color among all $36$disks and possesses the event “R and N”.

Now,

For the probability,

Number of favorable outcomes are divided by the number of possible outcomes.

$P(RandN)=\frac{Numberoffavarableoutcomes}{Numberofpossibleoutcomes}\approx 0.0278$

R: get a red disk

N: get a disk with the number$9$

For any two events,

General addition rule:

$P(AorB)=P\left(A\right)+P\left(B\right)-P(AorB)$

According to the statement,

$P(RorN)\ne P\left(R\right)+P\left(N\right)$

This is due to the events R and N are not mutually exclusive events.

That means

The occurrence of both the events at the same time is possible.

Such that

$P(RandN)\ne 0$

From Part (b) and (c),

We have

Probabilities,

For event R:

$P\left(R\right)=\frac{9}{36}$

For event N:

$P\left(N\right)=\frac{4}{36}$

For event “R and N”

$P(RandN)=\frac{1}{36}$

Apply general addition rule for any two events:

$$\begin{gathered} P(RorN)=P\left(R\right)+P\left(N\right)-P(RandN) \\ =\frac{9}{36}+\frac{4}{36}-\frac{1}{36} \\ =\frac{12}{36} \\ =\frac{1}{3} \\ \approx 0.3333 \end{gathered}$$