Chapter2

# Chapter2 - Chapter 2 Mathema-cal Probability ...

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Unformatted text preview: Chapter 2: Mathema-cal 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 Life-me of a light bulb Gene-cs: 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)  Iden-fy 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)  Iden-fy in words the event: {(x, x+1): x ∈ {1,2,3} } 3.  Life-me of a light bulb a)  Determine the event that the light bulb lasts between 100 and 110 hours. b)  Iden-fy in words the event: {x ∈ Ω: x < 200} 4.  Gene-cs: Dominant (A=black hair) or recessive (a = red hair) a)  Determine the event that the hair color is black. b)  Iden-fy 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: Probabili-es 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 distribu-on 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 Par--ons According to the Digest of Educa-on Sta-s-cs, 8.1% of the ins-tu-ons of higher educa-on 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. ins-tu-on of higher educa-on is selected at random, determine the probability that it is public. Example for Inclusion- Exclusion Principle Suppose that 80% of the popula-on read the Journal and Courier (J), 25% of the popula-on read the Exponent (E), 10% of the popula-on 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? ...
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