MATH 105: Finite Mathematics 7-2: Properties of Probability

Transcription

1 MATH 105: Finite Mathematics 7-2: Properties of Probability Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

2 Outline 1 Probability of Combined Events 2 Probability and Venn Diagrams 3 Odds 4 Conclusion

3 Outline 1 Probability of Combined Events 2 Probability and Venn Diagrams 3 Odds 4 Conclusion

4 Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a ball at random and note its color. 1 Find the probability of the event E =. 2 Find the probability of the simple event E = {G}. 3 Find the probability of the event E = {Green, Red}.

5 Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a ball at random and note its color. S = {R, G, B} 1 Find the probability of the event E =. 2 Find the probability of the simple event E = {G}. 3 Find the probability of the event E = {Green, Red}.

6 Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a ball at random and note its color. S = {R, G, B} 1 Find the probability of the event E =. 2 Find the probability of the simple event E = {G}. 3 Find the probability of the event E = {Green, Red}.

7 Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a ball at random and note its color. S = {R, G, B} 1 Find the probability of the event E =. 2 Find the probability of the simple event E = {G}. 3 Find the probability of the event E = {Green, Red}.

8 Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a ball at random and note its color. S = {R, G, B} 1 Find the probability of the event E =. 2 Find the probability of the simple event E = {G}. 3 Find the probability of the event E = {Green, Red}.

9 Unions of Events In the last part of the last problem, we could take E = {Green} and F = {Red} and note that: Union of Mutually Exclusive Events Let E and F be mutually exclusive (disjoint) events in a sample space S. Then, Pr[E F ] = Pr[E] + Pr[F ] You roll two fair six-sided dice and note the two numbers showing. With the sets E and F below, find Pr[E], Pr[F ], and Pr[E F ]. E = {(x, y) x + y is even } F = {(x, y) x + y 10}

10 Unions of Events In the last part of the last problem, we could take E = {Green} and F = {Red} and note that: Union of Mutually Exclusive Events Let E and F be mutually exclusive (disjoint) events in a sample space S. Then, Pr[E F ] = Pr[E] + Pr[F ] You roll two fair six-sided dice and note the two numbers showing. With the sets E and F below, find Pr[E], Pr[F ], and Pr[E F ]. E = {(x, y) x + y is even } F = {(x, y) x + y 10}

11 Unions of Events In the last part of the last problem, we could take E = {Green} and F = {Red} and note that: Union of Mutually Exclusive Events Let E and F be mutually exclusive (disjoint) events in a sample space S. Then, Pr[E F ] = Pr[E] + Pr[F ] You roll two fair six-sided dice and note the two numbers showing. With the sets E and F below, find Pr[E], Pr[F ], and Pr[E F ]. E = {(x, y) x + y is even } F = {(x, y) x + y 10}

12 General Unions General Unions of Events In general, if E and F are not-necessarily mutually exclusive events in a sample space S, then Pr[E F ] = Pr[E] + Pr[F ] Pr[E F ]

13 General Unions General Unions of Events In general, if E and F are not-necessarily mutually exclusive events in a sample space S, then Pr[E F ] = Pr[E] + Pr[F ] Pr[E F ] Recall that we used Venn Diagrams to help visualize this rule when it was stated for counting elements of sets. The same tool can be used for probability.

14 Outline 1 Probability of Combined Events 2 Probability and Venn Diagrams 3 Odds 4 Conclusion

15 Using Venn Diagrams Let A and B be events in a sample space S with Pr[A] = 0.60, Pr[B] = 0.40, and Pr[A B] = Use Venn Diagrams to find. 1 Pr[A B] 2 Pr[A] 3 Pr[A B] 4 Pr[A B]

16 Outline 1 Probability of Combined Events 2 Probability and Venn Diagrams 3 Odds 4 Conclusion

17 From Probability to Odds Many times probabilities are not expressed as numbers between 0 and 1, but rather as the odds for or against an event happening. Finding Odds If E is an event in a sample space S, then The odds for E are Pr[E] Pr[E] The odds against E are Pr[E] Pr[E] If the probability of an event E is 0.2, find the odds for and against the event.

18 From Probability to Odds Many times probabilities are not expressed as numbers between 0 and 1, but rather as the odds for or against an event happening. Finding Odds If E is an event in a sample space S, then The odds for E are Pr[E] Pr[E] The odds against E are Pr[E] Pr[E] If the probability of an event E is 0.2, find the odds for and against the event.

19 From Probability to Odds Many times probabilities are not expressed as numbers between 0 and 1, but rather as the odds for or against an event happening. Finding Odds If E is an event in a sample space S, then The odds for E are Pr[E] Pr[E] The odds against E are Pr[E] Pr[E] If the probability of an event E is 0.2, find the odds for and against the event.

20 From Odds to Probability You can also convert odds back into probability as shown below. Finding Probability If E is an event in a sample space S with the odds for E are a to b then Pr[E] = a a + b Pr[F ] = b a + b If the odds for A are 1 to 5 and the odds against B are 3 to 1, what are the odds for A or B assuming that A and B are mutually exclusive?

21 From Odds to Probability You can also convert odds back into probability as shown below. Finding Probability If E is an event in a sample space S with the odds for E are a to b then Pr[E] = a a + b Pr[F ] = b a + b If the odds for A are 1 to 5 and the odds against B are 3 to 1, what are the odds for A or B assuming that A and B are mutually exclusive?

22 From Odds to Probability You can also convert odds back into probability as shown below. Finding Probability If E is an event in a sample space S with the odds for E are a to b then Pr[E] = a a + b Pr[F ] = b a + b If the odds for A are 1 to 5 and the odds against B are 3 to 1, what are the odds for A or B assuming that A and B are mutually exclusive?

23 Outline 1 Probability of Combined Events 2 Probability and Venn Diagrams 3 Odds 4 Conclusion

24 Important Concepts Things to Remember from Section Events are just Sets, and can be treated as such. 2 Pr[A B] = Pr[A] + Pr[B] Pr[A B] 3 Conversion between probabilities and odds and back.

25 Important Concepts Things to Remember from Section Events are just Sets, and can be treated as such. 2 Pr[A B] = Pr[A] + Pr[B] Pr[A B] 3 Conversion between probabilities and odds and back.

26 Important Concepts Things to Remember from Section Events are just Sets, and can be treated as such. 2 Pr[A B] = Pr[A] + Pr[B] Pr[A B] 3 Conversion between probabilities and odds and back.

27 Important Concepts Things to Remember from Section Events are just Sets, and can be treated as such. 2 Pr[A B] = Pr[A] + Pr[B] Pr[A B] 3 Conversion between probabilities and odds and back.

28 Next Time... Now that we are hopefully more comfortable with probabilities, it is time to start using the counting rules we learned in the last chapter to help compute probabilities for more complicated events. In the next section we use Combinations and Permutations to compute probabilities. For next time Read Section 7-3 (pp ) Do Problem Sets 7-3 A,B

29 Next Time... Now that we are hopefully more comfortable with probabilities, it is time to start using the counting rules we learned in the last chapter to help compute probabilities for more complicated events. In the next section we use Combinations and Permutations to compute probabilities. For next time Read Section 7-3 (pp ) Do Problem Sets 7-3 A,B