THE MULTIPLICATION RULE AND INDEPENDENCE
Table of Contents
- 1. There are 50 students enrolled in the second year of a business degree program. During this semester, the students have to take some elective courses. 18 students are enrolled in psychology, 27 students are enrolled in geography, and 10 students are enrolled in both psychology and geography. A random student is to be selected.
- 2. The probability that the life of a product does NOT exceed t hours is 1 − e(−0.1*t). Given that a product has exceeded 10 hours, what is the probability it will fail in the next 5 hours?Links to an external site.
- 3. A word on P(A∩B)
- 4. The Multiplication Rule And Tree Diagrams
- 4.1. Example 1 : Suppose a baseball pitcher throws fastballs 80% of the time and curveballs 20% of the time. Suppose a batter hits a home run on 8% of all fastball pitches, and on 5% of all curveball pitches. What is the probability that this batter will hit a home run on this pitcher’s next pitch?
- 4.2. Example 2 : Suppose that 1.7% of people in a community have a particular disease. A particular test correctly identifies about 78% of people who truly have this disease. Also 10% of all tests result in false positives for patients who do not have this disease. If a test yields a positive result, what is the probability that patient actually has the disease?
1. There are 50 students enrolled in the second year of a business degree program. During this semester, the students have to take some elective courses. 18 students are enrolled in psychology, 27 students are enrolled in geography, and 10 students are enrolled in both psychology and geography. A random student is to be selected.
Discrete Data Trick
P(P) = 18/50
P(G) = 27/50
P(PnG) = 10/50
GC, P(PnGc) =8/50 -> G compliment the stuff thats not in G and in P
Trick
P(P) = P(PnG) + P(PnGc)
18/50 = 10/50 + 8/50
in how many ways can I get P?
2. The probability that the life of a product does NOT exceed t hours is 1 − e(−0.1*t). Given that a product has exceeded 10 hours, what is the probability it will fail in the next 5 hours?Links to an external site.
A = probability it will fail in next 5 hours B = product has exceeded 10 hours
P(A|B) = P(AnB)/P(B)
P(A) = 1-P(Ac) = .61
P(AnB) = P(A) - P(AnBc)
3. A word on P(A∩B)
3.1. Example 1 : Drawing two cards (without replacement)
A = {1st card king} B = {2nd card Queen}
P(AnB) is “the probability of drawing a King and then a Queen”
4. The Multiplication Rule And Tree Diagrams
AND then keyword is multiplication
multiplication rule
re: conditional probability P(B|A) = P(AnB)/P(A)
P(AnB) = P(A)*P(B|A)
for the king and queen from before
P(A) = 4/52 P(B) = 4/52
P(AnB) = P(A) * P(B|A) 4/52 * 4/51 = .006
A,B not independent!
4.1. Example 1 : Suppose a baseball pitcher throws fastballs 80% of the time and curveballs 20% of the time. Suppose a batter hits a home run on 8% of all fastball pitches, and on 5% of all curveball pitches. What is the probability that this batter will hit a home run on this pitcher’s next pitch?
f = {fast ball} c = {curve ball} h = {home run}
P(f) = 80/100
P(c) = 20/100
P(h|f) = 8/100
P(h|c) = 5/100
?P(h)?
Make a tree diagram for every possiblity and then add the two probabilities for homeruns to cacluate the P(H).
P(H) = P(CnH) + P(FnH) P(CnH) = .20 * .05 = .01 P(FnH) = .80 * .08 = .064 P(H) = .01 + .064 = .074
4.2. Example 2 : Suppose that 1.7% of people in a community have a particular disease. A particular test correctly identifies about 78% of people who truly have this disease. Also 10% of all tests result in false positives for patients who do not have this disease. If a test yields a positive result, what is the probability that patient actually has the disease?
D = {disease} P = {positive test)
P(D) = 1.7/100
P(P|D) = 78/100
P(P|D’) = 10/100
?P(D|P)?
P(D|P) = P(DnP)/P(P)
Probability of Disease and testing positive P(DnP) = P(D) * P(P|D) = .017 * .78 = .01333
Probability of Testing Poisitve P(P) = P(P|D’) + P(P|D) = .88 -> from diagram!
P(D|P) = .0133/.88 = .12
