MATH 105: Finite Mathematics 7-2: Properties of Probability Prof. - PowerPoint PPT Presentation

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion MATH 105: Finite Mathematics 7-2: Properties of Probability Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

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

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

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. Example 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 } .

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. Example 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 } .

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. Example 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 } .

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. Example 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 } .

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion Probabilities of Events Last time we looked at assigning probabilities to outcomes in a sample space. Example 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 } .

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 ] Example 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 }

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 ] Example 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 }

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 ] Example 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 }

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 ]

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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.

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

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion Using Venn Diagrams Example Let A and B be events in a sample space S with Pr [ A ] = 0 . 60, Pr [ B ] = 0 . 40, and Pr [ A ∩ B ] = 0 . 25. Use Venn Diagrams to find. 1 Pr [ A ∪ B ] 2 Pr [ A ] 3 Pr [ A ∩ B ] 4 Pr [ A ∪ B ]

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

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 ] Example If the probability of an event E is 0.2, find the odds for and against the event.

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 ] Example If the probability of an event E is 0.2, find the odds for and against the event.

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 ] Example If the probability of an event E is 0.2, find the odds for and against the event.

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 a b Pr [ E ] = Pr [ F ] = a + b a + b Example 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?

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 a b Pr [ E ] = Pr [ F ] = a + b a + b Example 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?

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion 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 a b Pr [ E ] = Pr [ F ] = a + b a + b Example 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?

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

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion Important Concepts Things to Remember from Section 7-2 1 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.

Probability of Combined Events Probability and Venn Diagrams Odds Conclusion Important Concepts Things to Remember from Section 7-2 1 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.