In: Statistics and Probability
An engineer has several pens in his desk (some are black and the rest are red). If he selects a pen at random, the probability it is a black pen equals .40. The proportion of black pens that are out of ink equals .34. The proportion of red pens that are out of ink equals .81. Draw a tree diagram and find the probabilities.
Group of answer choices
The probability a randomly selected pen is red.
[ Choose ] 0.6984 0.3780 0.8000 0.1620 0.4720 0.3310 0.1330 0.5660 0.4860 0.3733 0.6000 0.0380 0.6220 0.8475 0.5280 0.7814 0.3400 0.6136 0.7000 0.4000 0.8100 0.9329 0.2640 0.3240 0.6690 0.8390 0.2000 0.2040 0.3000 0.1900 0.1140 0.5670 0.6600 0.1020 0.3960 0.0760 0.2720 0.1980 0.1360 0.5982 0.4340
The probability a randomly selected pen is black and NOT out of ink.
[ Choose ] 0.6984 0.3780 0.8000 0.1620 0.4720 0.3310 0.1330 0.5660 0.4860 0.3733 0.6000 0.0380 0.6220 0.8475 0.5280 0.7814 0.3400 0.6136 0.7000 0.4000 0.8100 0.9329 0.2640 0.3240 0.6690 0.8390 0.2000 0.2040 0.3000 0.1900 0.1140 0.5670 0.6600 0.1020 0.3960 0.0760 0.2720 0.1980 0.1360 0.5982 0.4340
The probability a randomly selected pen is out of ink.
[ Choose ] 0.6984 0.3780 0.8000 0.1620 0.4720 0.3310 0.1330 0.5660 0.4860 0.3733 0.6000 0.0380 0.6220 0.8475 0.5280 0.7814 0.3400 0.6136 0.7000 0.4000 0.8100 0.9329 0.2640 0.3240 0.6690 0.8390 0.2000 0.2040 0.3000 0.1900 0.1140 0.5670 0.6600 0.1020 0.3960 0.0760 0.2720 0.1980 0.1360 0.5982 0.4340
The probability a randomly selected pen is red GIVEN it is out of ink.
[ Choose ] 0.6984 0.3780 0.8000 0.1620 0.4720 0.3310 0.1330 0.5660 0.4860 0.3733 0.6000 0.0380 0.6220 0.8475 0.5280 0.7814 0.3400 0.6136 0.7000 0.4000 0.8100 0.9329 0.2640 0.3240 0.6690 0.8390 0.2000 0.2040 0.3000 0.1900 0.1140 0.5670 0.6600 0.1020 0.3960 0.0760 0.2720 0.1980 0.1360 0.5982 0.4340
The probability a randomly selected pen is black GIVEN it is NOT out of ink.
Here is tree diagram for the selected pen , and if they are with/without ink.
Let R represent the Red Pen. , B represent Black Pen , I represent pen with Ink, W represent Pen without ink.
P(B) = 0.40 , P( Black pen out of Ink) = 0.34 , P ( Black pen with Ink) = 1 - 0.34 = 0.66
P(R)= 1-P(B) = 0.60 P(Red pen out of ink) = 0.81 , P(Red pen with Ink) = 1-0.81 = 0.19
1. From above tree diagram, The probability a randomly selected pen is red.
P(R) = 0.60
2. From above tree diagram and outcome probability table , The probability a randomly selected pen is black and NOT out of ink.
P(BI) = P(B) * P(I | B) =0.2640
3. The probability a randomly selected pen is out of ink.
Here is sample space = {RI,RW,BI,BW}
Outcome | Probability ( Outcome) |
RI | 0.60 *0.19 = 0.1140 |
RW | 0.60*0.81 = 0.486 |
BI | 0.40*0.66 = 0.264 |
BW | 0.40*0.34 = 0.136 |
P(Pen out of Ink) = P(RW) + P(BW) = 0.486 + 0.136 = 0.6220
4. The probability a randomly selected pen is red GIVEN it is out of ink. from Bayes theorem ,
P(R | W) =[ P(W|R)P(R) ] / [ P(W|R)P(R) + P(W|B)P(B) ] = P(RW) / [P(RW) + P(BW) ] = 0.486/(0.486+0.136) = 0.486/0.622 = 0.7813
5.The probability a randomly selected pen is black GIVEN it is NOT out of ink. from Bayes theorem
P(B | I) =[ P(I|B)P(B) ] / [ P(I|B)P(B) + P(I|R)P(R) ] = P(BI) / [P(BI) + P(RI) ] = 0.264/(0.264+0.114) = 0.264/0.378 = 0.6984