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1 1 Slide Introduction to Probability Probability Arithmetic and Conditional Probability Chapter 4 BA 201

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2 2 Slide PROBABILITY ARITHMETIC

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3 3 Slide The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. Addition Law The law is written as: P ( A B ) = P ( A ) + P ( B ) P ( A B

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4 4 Slide Event M = Markley Oil Profitable Event C = Collins Mining Profitable M C = Markley Oil Profitable or Collins Mining Profitable We know: P ( M ) = 0.70, P ( C ) = 0.48, P ( M C ) = 0.36 Thus: P ( M C) = P ( M ) + P( C ) P ( M C ) = 0.70 + 0.48 0.36 = 0.82 Addition Law (This result is the same as that obtained earlier using the definition of the probability of an event.) Bradley Investments

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5 5 Slide Scenario OutcomeProbability O1O1 0.10 O2O2 0.30 O3O3 0.05 O4O4 0.15 O5O5 0.20 O6O6 0.05 O7O7 0.10 O8O8 0.05 EventOutcomesProbability E1E1 O 1, O 3 0.15 E2E2 O 1, O 4, O 5, O 6 0.50 E3E3 O2O2 0.30 E4E4 O 7, O 8 0.15 E5E5 O 4, O 5, O 7 0.45 E6E6 O 1, O 7 0.20 A statistical experiment has the following outcomes, along with their probabilities and the following events, with the corresponding outcomes. Outcomes Events

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6 6 Slide Addition Law Using the addition law, what is the probability of E 2 ⋃ E 5 [P(E 2 ⋃ E 5 )]?

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7 7 Slide The probability of an event given that another event has occurred is called a conditional probability. A conditional probability is computed as follows : The conditional probability of A given B is denoted by P ( A | B ). Conditional Probability

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8 8 Slide Conditional Probability If you draw a card from a deck of cards, what is the probability of a Jack given you draw a face card, P(Jack|Face Card)? There are 12 face cards, 4 of which are Jacks. P(Jack|Face Card) = 4/12 = 1/3 Count and Divide P(Jack ⋂ Face Card) = 4/52 (since all Jacks are face cards) P(Face Card) = 12/52 P(Jack|Face Card) = P(Jack ⋂ Face Card) / P(Face Card) Computed = (4/52)/(12/52) = 1/3

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9 9 Slide Event M = Markley Oil Profitable Event C = Collins Mining Profitable We know: P ( M C ) = 0.36, P ( M ) = 0.70 Conditional Probability Thus: = Collins Mining Profitable given Markley Oil Profitable Bradley Investments

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10 Slide Conditional Probability What is the probability of E 5 given E 4 [P(E 5 | E 4 )]? P(E5 ⋂ E4)=0.10, P(E4)=0.15

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11 Slide Conditional Probability What is the probability of E 5 given E 4 [P(E 5 | E 4 )]? P(E5 ⋂ E4)=0.10, P(E4)=0.15 E4E5 O4, O5O7O8

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12 Slide Multiplication Law The multiplication law provides a way to compute the probability of the intersection of two events. The law is written as: P ( A B ) = P ( B ) P ( A | B ) P ( A B ) = P ( A ) P ( B | A ) OR

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13 Slide Event M = Markley Oil Profitable Event C = Collins Mining Profitable We know: P ( M ) = 0.70, P ( C | M ) = 0.5143 Multiplication Law M C = Markley Oil Profitable and Collins Mining Profitable Thus: P ( M C) = P ( M ) P ( M|C ) = (0.70)(0.5143) = 0.36 (This result is the same as that obtained earlier using the definition of the probability of an event.) Bradley Investments

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14 Slide Multiplication Law Using the multiplication rule, what is the probability of E 2 ⋂ E 5 [P(E 2 ⋂ E 5 )]? P(E2|E5)=0.7778, P(E5)=0.45 (OR) P(E5|E2)=0.70, P(E2)=0.50

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15 Slide Joint Probability Table Collins Mining Profitable (C) Not Profitable (C c ) Markley Oil Profitable (M) Not Profitable (M c ) Total 0.48 0.52 Total 0.70 0.30 1.00 0.36 0.34 0.12 0.18 Joint Probabilities (appear in the body of the table) Marginal Probabilities (appear in the margins of the table)

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16 Slide Joint Probability Table Collins Mining Profitable (C) Not Profitable (C c ) Markley Oil Profitable (M) Not Profitable (M c ) Total 0.48 0.52 Total 0.70 0.30 1.00 0.36 a 0.34 c 0.12 b 0.18 d P(M ⋂ C) = 0.20+0.16=0.36 a P(M c ⋂ C) = 0.10+0.02=0.12 b P(M ⋂ C c ) = 0.08+0.26=0.34 c P(M c ⋂ C c ) = 0.12+0.06=0.18 d

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17 Slide Independent Events If the probability of event A is not changed by the existence of event B, we would say that events A and B are independent. Two events A and B are independent if: P ( A | B ) = P ( A ) P ( B | A ) = P ( B ) or

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18 Slide Independent Events A bag contains three marbles, 1 blue and 2 red. If you draw a red marble, what is the probability the next marble you draw is blue? Blue = 1/3 Red= 2/3 Blue = 0 Red= 1 Blue = 1/2 Red= 1/2 P(Blue)=1/3 P(Blue|Red)=1/2 P(Red)=2/3 P(Red|Blue)=1

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19 Slide Independent Events If you flip a coin and get a head, what is the probability of getting a tail on the next flip? Head= 1/2 Tail=1/2 Head=1/2 Tail=1/2 Head=1/2 Tail=1/2 P(Head)=1/2 P(Head|Tail)=1/2 P(Tail)=1/2 P(Tail|Head)=1/2

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20 Slide The multiplication law also can be used as a test to see if two events are independent. The law is written as: P ( A B ) = P ( A ) P ( B ) Multiplication Law for Independent Events

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21 Slide Event M = Markley Oil Profitable Event C = Collins Mining Profitable We know: P ( M C ) = 0.36, P ( M ) = 0.70, P ( C ) = 0.48 But: P ( M)P(C) = (0.70)(0.48) = 0.34, not 0.36 Are events M and C independent? Does P ( M C ) = P ( M)P(C) ? Hence: M and C are not independent. Bradley Investments Multiplication Law for Independent Events

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22 Slide Multiplication Law for Independent Events E h1 =Head on first flip. E h2 =Head on second flip. P(E h1 ⋂ E h2 ) Head= 1/2 Tail=1/2 Head=1/2 Tail=1/2 Head=1/2 Tail=1/2 H, H=1/4=P(E h1 ⋂ E h2 ) P(E h1 )=1/2 P(E h2 )=1/2 P(E h1 ) P(E h2 )=1/4 P(E h1 ⋂ E h2 )=1/4 P(E h1 ⋂ E h2 )= P(E h1 ) P(E h2 )

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23 Slide Do not confuse the notion of mutually exclusive events with that of independent events. Two events with nonzero probabilities cannot be both mutually exclusive and independent. If one mutually exclusive event is known to occur, the other cannot occur; thus, the probability of the other event occurring is reduced to zero (and they are therefore dependent ). Mutual Exclusiveness and Independence Two events that are not mutually exclusive, might or might not be independent.

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