Bayes Theorem

[nnk]

Hello, I have a set of problems using Bayes Theorem and one using the Law of Total Probability. I have worked through the problems, but do not know if I have the correct answers. I have listed the problems below with the methods I used to get my answers. I would greatly appreciate if someone can assist and confirm these are correct answers. Thanks!

1. Suppose there are two bowls of candy. Bowl #1 has 10 red M&Ms and 30 brown M&Ms, while bowl #2 has 20 of each color. You pick a bowl at random and pick a single piece of candy at random.
   1. If you pick a brown M&M, what is the probability that it came from bowl #1?

P(Bowl #1 | Brown)= .75(probability of picking brown from bowl #1) * .5 (probability of choosing bowl 1)
______________________________________________________________________________________________________________ = .6
(probability of choosing brown) .625

2. If you pick a brown M&M, what is the probability that it came from bowl#2?

P(Bowl #2 | Brown)= .5(probability of picking brown from bowl #2) * .5 (probability of choosing bowl 1)
______________________________________________________________________________________________________________ = .4
(probability of choosing brown) .625


2. At a college, 4% of male students are taller than 6 feet and 1% of female students are taller than 6 feet tall. The ratio of female to male students is 3:2 in favor of women. If a student is selected at random from among all those who are taller than six feet, what is the probability that the student is a woman?

P(a woman |over 6 ft. tall)= .01 (probability of choosing a woman over 6ft)*.6 (probability of choosing a woman)
______________________________________________________________________________________________________________________ = .27
.022(probability of selecting someone over 6ft)

3. A factory manufactures bolts using three machines, A, B and C. Out of these, A produces for 25% of the total, B produces 35% of the total and C produces the rest. From earlier experience, we know that A produces 5% defective bolts, B produces 4% defective ones and C produces 2% defective. A randomly chosen bolt from the production is found to be defective. What is the probability that it came from:
(i) machine A:
P(Machine A | defective bolt)= .05(probability that Machine A makes a defective bolt)*.25(probability of it coming from A)
__________________________________________________________________________________________________________________________________ = .36
.0345 (probability that there is a defective bolt)

(ii) machine B
P(Machine B | defective bolt)=.04(probability that Machine B makes a defective bolt)*.35(probability of it coming from B)
__________________________________________________________________________________________________________________________________ = .13
.0345 (probability that there is a defective bolt)

(iii) machine C
P(Machine C | defective bolt)=.02(probability that Machine C makes a defective bolt)*.4(probability of it coming from C)
__________________________________________________________________________________________________________________________________ = .23
.0345 (probability that there is a defective bolt)

4. Suppose that we have two identical boxes: box 1 and box 2. Box 1 contains 5 red balls and 3 blue balls. Box 2 contains 2 red balls and 4 blue balls. A box is selected at random and exactly one ball is drawn from the box.

(i) Using the law of total probability, what is the probability that the ball is blue?
P(ball is blue)= P(blue|box1) .375 * P(box1) .5 + P(blue|box2) .667 * P(box2) .5

.5(.375) + .5(.667) = .52

(iI) Given that the selected ball is blue, what’s the probability that it came from box 2?
P (came from box 2| ball is blue)= .667 (probability that the blue ball came from box 2) * .5(probability of it coming from box 2)
_______________________________________________________________________________________________________________________________________ = .32
.667+.375 (probability that ball is blue)


Thank You!!

 

[pka]

Hello, I have a set of problems using Bayes Theorem and one using the Law of Total Probability. I have worked through the problems, but do not know if I have the correct answers. I have listed the problems below with the methods I used to get my answers. I would greatly appreciate if someone can assist and confirm these are correct answers. Thanks!

1. Suppose there are two bowls of candy. Bowl #1 has 10 red M&Ms and 30 brown M&Ms, while bowl #2 has 20 of each color. You pick a bowl at random and pick a single piece of candy at random.
   1. If you pick a brown M&M, what is the probability that it came from bowl #1?

P(Bowl #1 | Brown)= .75(probability of picking brown from bowl #1) * .5 (probability of choosing bowl 1)
______________________________________________________________________________________________________________ = .6
(probability of choosing brown) .625

2. If you pick a brown M&M, what is the probability that it came from bowl#2?

P(Bowl #2 | Brown)= .5(probability of picking brown from bowl #2) * .5 (probability of choosing bowl 1)
______________________________________________________________________________________________________________ = .4
(probability of choosing brown) .625


2. At a college, 4% of male students are taller than 6 feet and 1% of female students are taller than 6 feet tall. The ratio of female to male students is 3:2 in favor of women. If a student is selected at random from among all those who are taller than six feet, what is the probability that the student is a woman?

P(a woman |over 6 ft. tall)= .01 (probability of choosing a woman over 6ft)*.6 (probability of choosing a woman)
______________________________________________________________________________________________________________________ = .27
.022(probability of selecting someone over 6ft)

3. A factory manufactures bolts using three machines, A, B and C. Out of these, A produces for 25% of the total, B produces 35% of the total and C produces the rest. From earlier experience, we know that A produces 5% defective bolts, B produces 4% defective ones and C produces 2% defective. A randomly chosen bolt from the production is found to be defective. What is the probability that it came from:
(i) machine A:
P(Machine A | defective bolt)= .05(probability that Machine A makes a defective bolt)*.25(probability of it coming from A)
__________________________________________________________________________________________________________________________________ = .36
.0345 (probability that there is a defective bolt)

(ii) machine B
P(Machine B | defective bolt)=.04(probability that Machine B makes a defective bolt)*.35(probability of it coming from B)
__________________________________________________________________________________________________________________________________ = .13
.0345 (probability that there is a defective bolt)

(iii) machine C
P(Machine C | defective bolt)=.02(probability that Machine C makes a defective bolt)*.4(probability of it coming from C)
__________________________________________________________________________________________________________________________________ = .23
.0345 (probability that there is a defective bolt)

4. Suppose that we have two identical boxes: box 1 and box 2. Box 1 contains 5 red balls and 3 blue balls. Box 2 contains 2 red balls and 4 blue balls. A box is selected at random and exactly one ball is drawn from the box.

(i) Using the law of total probability, what is the probability that the ball is blue?
P(ball is blue)= P(blue|box1) .375 * P(box1) .5 + P(blue|box2) .667 * P(box2) .5

.5(.375) + .5(.667) = .52

(iI) Given that the selected ball is blue, what’s the probability that it came from box 2?
P (came from box 2| ball is blue)= .667 (probability that the blue ball came from box 2) * .5(probability of it coming from box 2)
_______________________________________________________________________________________________________________________________________ = .32
.667+.375 (probability that ball is blue)


Thank You!!

Hello, I have a set of problems using Bayes Theorem and one using the Law of Total Probability. I have worked through the problems, but do not know if I have the correct answers. I have listed the problems below with the methods I used to get my answers. I would greatly appreciate if someone can assist and confirm these are correct answers. Thanks!

1. Suppose there are two bowls of candy. Bowl #1 has 10 red M&Ms and 30 brown M&Ms, while bowl #2 has 20 of each color. You pick a bowl at random and pick a single piece of candy at random.
   1. If you pick a brown M&M, what is the probability that it came from bowl #1?

P(Bowl #1 | Brown)= .75(probability of picking brown from bowl #1) * .5 (probability of choosing bowl 1)
______________________________________________________________________________________________________________ = .6
(probability of choosing brown) .625

2. If you pick a brown M&M, what is the probability that it came from bowl#2?

P(Bowl #2 | Brown)= .5(probability of picking brown from bowl #2) * .5 (probability of choosing bowl 1)
______________________________________________________________________________________________________________ = .4
(probability of choosing brown) .625


2. At a college, 4% of male students are taller than 6 feet and 1% of female students are taller than 6 feet tall. The ratio of female to male students is 3:2 in favor of women. If a student is selected at random from among all those who are taller than six feet, what is the probability that the student is a woman?

P(a woman |over 6 ft. tall)= .01 (probability of choosing a woman over 6ft)*.6 (probability of choosing a woman)
______________________________________________________________________________________________________________________ = .27
.022(probability of selecting someone over 6ft)

3. A factory manufactures bolts using three machines, A, B and C. Out of these, A produces for 25% of the total, B produces 35% of the total and C produces the rest. From earlier experience, we know that A produces 5% defective bolts, B produces 4% defective ones and C produces 2% defective. A randomly chosen bolt from the production is found to be defective. What is the probability that it came from:
(i) machine A:
P(Machine A | defective bolt)= .05(probability that Machine A makes a defective bolt)*.25(probability of it coming from A)
__________________________________________________________________________________________________________________________________ = .36
.0345 (probability that there is a defective bolt)

(ii) machine B
P(Machine B | defective bolt)=.04(probability that Machine B makes a defective bolt)*.35(probability of it coming from B)
__________________________________________________________________________________________________________________________________ = .13
.0345 (probability that there is a defective bolt)

(iii) machine C
P(Machine C | defective bolt)=.02(probability that Machine C makes a defective bolt)*.4(probability of it coming from C)
__________________________________________________________________________________________________________________________________ = .23
.0345 (probability that there is a defective bolt)

4. Suppose that we have two identical boxes: box 1 and box 2. Box 1 contains 5 red balls and 3 blue balls. Box 2 contains 2 red balls and 4 blue balls. A box is selected at random and exactly one ball is drawn from the box.

(i) Using the law of total probability, what is the probability that the ball is blue?
P(ball is blue)= P(blue|box1) .375 * P(box1) .5 + P(blue|box2) .667 * P(box2) .5

.5(.375) + .5(.667) = .52

(iI) Given that the selected ball is blue, what’s the probability that it came from box 2?
P (came from box 2| ball is blue)= .667 (probability that the blue ball came from box 2) * .5(probability of it coming from box 2)
_______________________________________________________________________________________________________________________________________ = .32

You have posted a whole laundry list of questions. We do not function as a checking service.
Here is the setup for #1. $\displaystyle \mathscr{P}(1|B)=\dfrac{\mathscr{P}(B|1)\mathscr{P}(1)}{\mathscr{P}(B|1)\mathscr{P}(1)+\mathscr{P}(B|2)\mathscr{P}(2)}$

Here is the setup for #3
$\displaystyle \mathscr{P}(A|D)=\frac{\mathscr{P}(D|A)\mathscr{P}(A)}{\mathscr{P}(D|A)\mathscr{P}(A)+\mathscr{P}(D|B)\mathscr{P}(B)+\mathscr{P}(D|C)\mathscr{P}(C)}$

 

[nnk]

You have posted a whole laundry list of questions. We do not function as a checking service.
Here is the setup for #1. $\displaystyle \mathscr{P}(1|B)=\dfrac{\mathscr{P}(B|1)\mathscr{P}(1)}{\mathscr{P}(B|1)\mathscr{P}(1)+\mathscr{P}(B|2)\mathscr{P}(2)}$

Here is the setup for #3
$\displaystyle \mathscr{P}(A|D)=\frac{\mathscr{P}(D|A)\mathscr{P}(A)}{\mathscr{P}(D|A)\mathscr{P}(A)+\mathscr{P}(D|B)\mathscr{P}(B)+\mathscr{P}(D|C)\mathscr{P}(C)}$

My apologies for the long post. Thank you for your answer, it was very helpful.