Probability-solutions

Probability-solutions - Introduction to probability Note...

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Introduction to probability Note that a formal/rigorous introduction to probability is beyond the scope of this course. Probability is a numerical measure of uncertainty. Uncertainties are abundant. Consider uncertainties about the weather, a medical diagnosis, success of a marriage, yield of a corn crop, the consequences of any decision etc. Examples: A weather forecaster asserts that there is a 60% chance of snow today A physician asserts that there is a 95% chance that an operation will be successfull An engineer claims that a 6-sided die will land on “3” with probability 1/6. We will use the basic concepts of probability theory, in conjunction with knowledge about a population or process, in order to specify a model for observations from that population or process. 1 Experiments, outcomes, and events 1.1 Definitions Definition: experiment action or process that generates outcomes. Only one outcome can occur and we are usually uncertain which outcome this will be. Definition: sample space set of all outcomes of an experiment. Denoted by S . We will now discuss sample spaces for which we can enumerate the outcomes. Example 1.1: Specify the sample space for the following experiments 1. Toss a coin: S = { H, T } 2. Toss two coins: S = { HH, HT, TH, TT } 3. Toss a coin until the first “heads”: S = { H, TH, TTH, TTTH, . . . } 4. Roll one 6-sided die: S = { 1 , 2 , 3 , 4 , 5 , 6 } 1
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5. Roll two 6-sided dice: S = { (1 , 1) , (1 , 2) , . . . , (6 , 6) } Definition: event any subset of outcomes (any subset of S ) an event occurs if the occurring outcome is in the event. an event may have zero, one, many, or all outcomes of the experiment. Example 1.2: Experiment: test if two batteries work Let W = “works” and let F = “fails”. S = { WW, WF, FW, FF } Specify the outcomes in the following events: 1. Event A : “the first battery works” A = { WW, WF } 2. Event B : “at least one works” B = { WW, WF, FW } 3. Event C : “exactly one works” C = { WF, FW } 4. Event D : “both batteries fail” D = { FF } 1.2 Events are sets For any experiment with sample space S and events A and B . Definition: Union, A B the event consisting of all outcomes in either A or B or both. Note that A B = B A . Definition: Intersection, A B the event consisting of all outcomes in both A and B . Note that A B = B A . 2
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Definition: Compliment, A c the event consisting of all outcomes in S that are not in A . Venn diagrams: Definition: disjoint events When events A and B have no outcomes in common, denoted A B = , we say A and B are disjoint events. Definition: null event denoted by , an event with zero outcomes. It cannot occur. Note: All outcomes in the sample space S are disjoint events. Example 1.3: Experiment: flip two coins. S = { HH, HT, TH, TT } Let A =“Heads on first flip” and B =“Tails on first flip”, then A = { HH, HT } and B = { TH, TT } thus A B = Example 1.4: Experiment: roll one 6-sided die S = { 1 , 2 , 3 , 4 , 5 , 6 } Specify the outcomes in each of the following events 1. A = “the roll is less than 4” A = { 1 , 2 , 3 } 2. B = “the die shows 3” B = { 3 } 3. C = “the die shows an even number” C = { 2 , 4 , 6 } 4. A B = { 3 } 5. B C = { 2 , 3 , 4 , 6 } 6. B C c = { 3 } 7. B C = 3
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