Unformatted text preview: Chapter 2: Mathemacal Probability h5p://www.cartoonstock.com/directory/p/probability.asp Sample Spaces: Examples 1.
2.
3.
4. Tossing Coins: We toss a coin 3 mes Rolling two 4 sided dies Lifeme of a light bulb Genecs: Dominant (A=black hair) or recessive (a = red hair) Events: Examples 1. Tossing Coins: 3 mes a) Determine the event that there is only one Head. b) Idenfy in words the event: {HHH,TTT} 2. Rolling two 4 sided dies a) Determine the event that the sum of the two dice is 9. b) Determine the event that the diﬀerence between the numbers of the white and red dies is 1. c) Idenfy in words the event: {(x, x+1): x ∈ {1,2,3} } 3. Lifeme of a light bulb a) Determine the event that the light bulb lasts between 100 and 110 hours. b) Idenfy in words the event: {x ∈ Ω: x < 200} 4. Genecs: Dominant (A=black hair) or recessive (a = red hair) a) Determine the event that the hair color is black. b) Idenfy in words the event: {aa, aA, Aa} Subsets and Events: Example Rolling 2 4 sided dice: A: the red die is a 4 B: The sum of the two dice is 4 a) AC b) A ∩ B c) A U B d) AC ∩ B e) AC U B Mutually Exclusive Events: Example Rolling 2 4 sided dice: Events: A: the event that you roll a 2 on the red die B: the event that you roll a 3 on the white die C: the event that the sum of the dice is 3 D: the event that the sum of the dice is 2 Which of the following are mutually exclusive? a) A and B b) A and C c) A and D d) B and C e) B and D f) C and D g) A, B and C h) B, C and D Example: Probabilies for Countable Sample Spaces Row # Outcome #1 #2 #3 #4 #5 #6 1 11 0.0625 0 0.1 0.1 0.05 0.05 2 12 0.0625 0 0.1 0.1 0.05 0.05 3 13 0.0625 0 0.1 0.1 0.05 0.05 4 14 0.0625 0 0.1 0.1 0.05 0.05 5 21 0.0625 0.25 0.1 0.2 0.1 0.1 6 22 0.0625 0.25 0.1 0.2 0.1 0.1 7 23 0.0625 0.25 0.1 0.2 0.1 0.1 8 24 0.0625 0.25 0.1 0.2 0.1 0.1 9 31 0.0625 0 0.1 0.1 0 0.01 10 32 0.0625 0 0.1 0.1 0 0.01 11 33 0.0625 0 0.1 0.1 0 0.01 12 34 0.0625 0 0.1 0.1 0 0.01 13 41 0.0625 0 0.1 0.05 0.1 0.05 14 42 0.0625 0 0.1 0.05 0.1 0.05 15 43 0.0625 0 0.1 0.05 0.1 0.05 16 44 0.0625 0 0.1 0.05 0.1 0.05 Table 2.5: Age and rank of faculty members Rank Full
Associate
Professor Professor
R1
R2 Assistant
Professor
R3 Instructor
R4 Total 2 3 57 6 68 30  39
A2
Age Under 30
A1 52 170 163 17 402 40  49
A3 156 125 61 6 348 50  59
A4 145 68 36 4 253 60 & older
A5 75 15 3 0 93 Total 430 381 330 33 1164 Table 2.6: Joint probability distribuon corresponding to Table 2.5 Rank
Full
Associate
Professor Professor
R1
R2 Assistant
Professor
R3 Instructor
R4 P(Aj) 0.002 0.003 0.049 0.005 0.058 30  39
A2
Age Under 30
A1 0.045 0.146 0.140 0.015 0.345 40 49
A3 0.134 0.107 0.052 0.005 0.299 50  59
A4 0.125 0.058 0.031 0.003 0.217 60 & older
A5 0.064 0.013 0.003 0.000 0.080 P(Rj) 0.369 0.327 0.275 0.028 1.000 Example 2.28: Law of Parons According to the Digest of Educaon Stascs, 8.1% of the instuons of higher educaon are public schools in the Northeast, 11.0% are public schools in the Midwest, 16.3% are public schools in the South, and 9.6% are public schools in the West. If a U.S. instuon of higher educaon is selected at random, determine the probability that it is public. Example for Inclusion Exclusion Principle Suppose that 80% of the populaon read the Journal and Courier (J), 25% of the populaon read the Exponent (E), 10% of the populaon read the New York Times (N), 15% read both the J and the E, 5% read both the J and the N, 2% read both E and the N and 1% read all three. If a person from this city is selected at random, what is the probability that she does not read at least one of the newspapers? ...
View
Full Document
 Spring '08
 Staff
 Probability, Dice, Privatdozent, Docent

Click to edit the document details