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Chapter 5: Q. 46 (page 328)

Colorful disksA jar contains 36disks: 9each of four colors鈥攔ed, green, blue, and Page Number: 328yellow. Each set of disks of the same color is numbered from 1to 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 Rand N.

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

c. Describe the event 鈥Rand N鈥 in words. Then find the probability of this event.

d. Explain why P(RorN)鈮P(R)+P(N) Then use the general addition rule to computeP(RorN).

Short Answer

Expert verified

a. A two 鈭 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 Rand Nis0.1111

c. The probability of "RandN" is0.0278

d. The probability of "RorN" is0.3333

Step by step solution

01

Part (a) Step 1 : Given Information

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

02

Part (a) Step 2 : Simplification

R: grab yourself a red disc
Get a disc with the number 9on 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 Rand Nevents.

Rows : red , not red

Columns : nine , not nine
We must write 36in the bottom left corner of the table because there are 36discs in total.
We are aware of this.
There are 9red, blue, green, and yellow discs in total.
This denotes
The discs are crimson in hue and there are nine of them and the remaining 27blue, green, and yellow discs.
Wealsounderstandthat
Each colour disc is assigned a number from 1to 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 9and is tinted red; the remaining eight discs are numbered 1-8.
Due to the fact that 3discs of different colours are numbered 9, the remaining 24discs of different colours are numbered 1-8.

Two 鈭 way table is shown as:


 Nine Not Nine Total
Red 1 8 9
Not red 3 24 27
Total 4
 32 36
03

Part (b) Step 1 : Given Information

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

04

Part (b) Step 2 : Simplification

R:grabyourselfareddisk
Get a disk with the number 9on it (N).
We are aware of this.
The colour red appears on 9of the 36discs.
The probability is calculated by dividing the number of favourable outcomes by the total number of possible possibilities.

P(R)=Number鈥of鈥favourable鈥outcomesNumber鈥of鈥possible鈥outcomes=936=14=0.25

Wealsounderstandthat
Each colour disc is assigned a number from 1to 9.
4discs with 4colours are numbered 9out of 36discs.

P(N)=Number鈥of鈥favourable鈥outcomesNumber鈥of鈥possible鈥outcomes=436=19鈮0.1111

05

Part (c) Step 1 : Given Information

We have to discuss the event 鈥R and N鈥 and also find its probability.

06

Part (c) Step 2 : Simplification

R: grab yourself a red disc
Get a disc with the number 9on it (N).
The letters "Rand N" stand for the fact that the disc is numbered 9and is red in hue.
We are aware of this.
There are 9red discs in all, out of a total of 36discs.
Since each colour disc is numbered 1to 9, there are a total of 9colour discs.
This denotes
Among the 36discs, there is only one that is numbered 9and red in colour and has the event "Rand N."
Now,
In terms of the likelihood,
The number of favourable results is divided by the total number of options.

P(R鈥and鈥N)=Number鈥of鈥favourable鈥outcomesNumber鈥of鈥possible鈥outcomes鈮0.0278

07

Part (d) Step 1 : Given Information

We have to discuss the probability of event 鈥RorN鈥 is not equal to the sum of individual probabilities of both events. Furthermore, general addition rule to be used for the probability of event 鈥RorN鈥.

08

Part (d) Step 2 : Simplification

R: grab yourself a red disc
Get a disc with the number 9on it (N).
If there are two events,
Rule of thumb for addition:

P(A鈥or鈥B)=P(A)+P(B)鈭P(A鈥and鈥B)

According to the declaration,

P(R鈥or鈥N)鈮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鈥and鈥N)鈮0

Parts (b) and (c),
We've got
Probabilities,
For the Revent: P(R)=936

For event N: 鈥鈥P(N)=436

For event 鈥RandN鈥 :- P(R鈥and鈥N)=136

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

P(R鈥or鈥N)=P(R)+P(N)鈭P(R鈥and鈥N)鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥=936+436鈭136鈥鈥鈥鈥鈥鈥=1236鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈮0.3333

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