Chapter 5: Q. 46 (page 328)

Colorful disksA jar contains $36$ disks: $9$ each of four colors—red, green, blue, and Page Number: $328$ yellow. Each set of disks of the same color is numbered from $1$ to $9$ . 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 $P (R o r N) \neq P (R) + P (N)$ Then use the general addition rule to compute $P (R o r N) .$

Short Answer

Expert verified

a. A two − way table describing sample space in terms of events $R$ and $N$ .

	Nine	Not Nine	Total
Red	$1$	$8$	$9$
Not red	$3$	$24$	$27$
Total	$4$	$32$	$36$

b. The probabilities of both events $R$ and $N$ is $0.1111$

c. The probability of " $R a n d N$ " is $0.0278$

d. The probability of " $R o r N$ " is $0.3333$

Step by step solution

Part (a) Step 1 : Given Information

We have to determine two − way table describing sample space in terms of events $R$ and $N$ .

Part (a) Step 2 : Simplification

R

: grab yourself a red disc

Get a disc with the number

9

on it

N

In the case of a two-way table,
It is necessary for us to create two rows and two columns.
Let's call the rows and columns after the

R

and

N

events.

Rows : red , not red

Columns : nine , not nine

We must write

36

in the bottom left corner of the table because there are

36

discs in total.

We are aware of this.
There are

9

red, blue, green, and yellow discs in total.

This denotes

The discs are crimson in hue and there are nine of them and the remaining

27

blue, green, and yellow discs.

Wealsounderstandthat

Each colour disc is assigned a number from

1

9

This denotes

There is only one disc, which is crimson in colour and has the number nine on it.

And

The last three discs are numbered 9 and come in a variety of colours.

This suggests that

Only one disc is numbered

9

and is tinted red; the remaining eight discs are numbered

1 - 8

Due to the fact that

3

discs of different colours are numbered 9, the remaining

24

discs of different colours are numbered

1 - 8

Two − way table is shown as:

	Nine	Not Nine	Total
Red	$1$	$8$	$9$
Not red	$3$	$24$	$27$
Total	$4$	$32$	$36$

Part (b) Step 1 : Given Information

We have to determine the probabilities of both events $R$ and $N$ .

Part (b) Step 2 : Simplification

R:grabyourselfareddisk

Get a disk with the number

9

on it (

N

We are aware of this.
The colour red appears on

9

of the

36

discs.

The probability is calculated by dividing the number of favourable outcomes by the total number of possible possibilities.

$P (R) = \frac{N u m b e r o f f a v o u r a b l e o u t c o m e s}{N u m b e r o f p o s s i b l e o u t c o m e s} = \frac{9}{36} = \frac{1}{4} = 0.25$

Wealsounderstandthat

Each colour disc is assigned a number from

1

9

4

discs with

4

colours are numbered

9

out of

36

discs.

$P (N) = \frac{N u m b e r o f f a v o u r a b l e o u t c o m e s}{N u m b e r o f p o s s i b l e o u t c o m e s} = \frac{4}{36} = \frac{1}{9} \approx 0.1111$

Part (c) Step 1 : Given Information

We have to discuss the event “ $R$ and $N$ ” and also find its probability.

Part (c) Step 2 : Simplification

R

: grab yourself a red disc

Get a disc with the number

9

on it (

N

The letters "

R

and

N

" stand for the fact that the disc is numbered

9

and is red in hue.

We are aware of this.

There are

9

red discs in all, out of a total of

36

discs.

Since each colour disc is numbered

1

9

, there are a total of

9

colour discs.

This denotes

Among the

36

discs, there is only one that is numbered

9

and red in colour and has the event "

R

and

N

Now,

In terms of the likelihood,

The number of favourable results is divided by the total number of options.

$P (R a n d N) = \frac{N u m b e r o f f a v o u r a b l e o u t c o m e s}{N u m b e r o f p o s s i b l e o u t c o m e s} \approx 0.0278$

Part (d) Step 1 : Given Information

We have to discuss the probability of event “ $R o r N$ ” is not equal to the sum of individual probabilities of both events. Furthermore, general addition rule to be used for the probability of event “ $R o r N$ ”.

Part (d) Step 2 : Simplification

R

: grab yourself a red disc

Get a disc with the number

9

on it (

N

If there are two events,

Rule of thumb for addition:

$P (A o r B) = P (A) + P (B) - P (A a n d B)$

According to the declaration,

$P (R o r N) \neq P (R) + P (N)$

ThereasonforthisisthattheoccurrencesRandNarenotmutuallyexclusive.

That is to say,

It is feasible for both occurrences to occur at the same moment.

as a result

$P (R a n d N) \neq 0$

Parts (

b

) and (

c

We've got

Probabilities,

For the

R

event:

P (R) = \frac{9}{36}

For event N: $P (N) = \frac{4}{36}$

For event “ $R a n d N$ ” :- $P (R a n d N) = \frac{1}{36}$

For any two events, use the following generic addition rule:

$P (R o r N) = P (R) + P (N) - P (R a n d N) = \frac{9}{36} + \frac{4}{36} - \frac{1}{36} = \frac{12}{36} \approx 0.3333$

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Short Answer

Step by step solution

Part (a) Step 1 : Given Information

Part (a) Step 2 : Simplification

Part (b) Step 1 : Given Information

Part (b) Step 2 : Simplification

Part (c) Step 1 : Given Information

Part (c) Step 2 : Simplification

Part (d) Step 1 : Given Information

Part (d) Step 2 : Simplification

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