P(B A) P(A And B) P(A)P(B A) Or P(B A) P( A And B) P(A)

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Brent ClaytonHomework #0Summary of class notes 2/3/2011The lecture notes for this particular day a centered on sections 2.4-2.5 in the Devore book.Basically the idea of conditional probability is presented here. Conditional probability statesthat given an event (event B), it will only occur if an event (event A) has already occurred. Theprobabilty statement can be written as follows:P(B A)This only applies when the events are independent of each other meaning event A has no effecton the probability of event B happening. The other case involes these two events when theyare independent. This scenario produces an intersection of the two events (the probability thatboth events occur). This can be written as follows:P(A and B) P(A)P(B A) orP(B A) P( A and B)P(A)Example: (Deck of Cards)B Red CardPr(A1) .25 (Probability the card is a diamond)A(1) Card is a DiamondPr(A1 B) .50 (Probability that the card is aDiamond given it is Red)A(2) Card is a JackPr(A2 B) 1/13 (Probability the card is a Jack ofDiamond given it is Red)

This brings us to the subject of mutually exclusive events. The definition of being mutuallyexclusive (disjoint) means that it is impossible for two events to occur together. Given twoevents, A and B, they are mutually exclusive if (A П B) 0. If these two events are mutuallyexclusive, they cannot be independent.Question: If A, B are mutually exclusive, then A, B are independent ?a) Sometimesb) Alwaysc) NeverCorrect answer is c.Example (Mutually Exclusive Events):A recent study was conducted using both male and female subjects ages 20-30 that wanted tofind the average salary of men vs women.For the example here, the mutually exclusive events are the subject in the study could not beboth female and male at the same time. The subject also cannot also be aged 22 or 28 (randomselection, any of the ages would work in the range given) at the same time.

Brian HendersonST 371 Homework 0Lecture of 03FEB2011This lecture summary covers parts of conditional probability. We went through several examples of how todetermine the different probabilities using a Venn diagram. We covered also the difference between independent anddependent events.Conditional Probability – the probability that one event will occur given that another event has already occurred.Notation: Probability that A will occur given B has occurred already Pr(A B).Pr(𝐴 𝐵) Pr(𝐴 𝐵)Pr(𝐵)Events are independent if Pr(A B) Pr(A) and are dependent other wise.Example. Given a Normal Deck of Playing Cards.A Diamond Card B Red Card C JackExample 1)11Pr(𝐴 𝐵) 4 50%1 22 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑟𝑑 𝑏𝑒𝑖𝑛𝑔 𝑎 𝑑𝑖𝑎𝑚𝑜𝑛𝑑 𝑎𝑓𝑡𝑒𝑟 𝑘𝑛𝑜𝑤𝑖𝑛𝑔 𝑡ℎ𝑎𝑡 𝑖𝑡 𝑖𝑠 𝑎 𝑟𝑒𝑑 𝑐𝑎𝑟𝑑13 1Pr(𝐴) 25% 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑟𝑑 𝑏𝑒𝑖𝑛𝑔 𝑎 𝑑𝑖𝑎𝑚𝑜𝑛𝑑 𝑐𝑎𝑟𝑑52 4 Pr(𝐴 𝐵) Pr(𝐴) 𝑤ℎ𝑖𝑐ℎ 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑒𝑣𝑒𝑛𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑛 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟.Example 2) Using same event designations as example 1.2152Pr(𝐶 𝐵) 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑟𝑑 𝑏𝑒𝑖𝑛𝑔 𝑎 𝐽𝑎𝑐𝑘 𝑎𝑓𝑡𝑒𝑟 𝑘𝑛𝑜𝑤𝑖𝑛𝑔 𝑡ℎ𝑎𝑡 𝑖𝑡 𝑖𝑠 𝑎 𝑟𝑒𝑑 𝑐𝑎𝑟𝑑113241Pr(𝐶) 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑟𝑑 𝑏𝑒𝑖𝑛𝑔 𝑎 𝐽𝑎𝑐𝑘52 13 Pr(𝐶 𝐵) Pr(𝐶) 𝑤ℎ𝑖𝑐ℎ 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑒𝑣𝑒𝑛𝑡𝑠 𝐵 𝑎𝑛𝑑 𝐶 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑛 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟.If A is independent of B, then A’ is also independent of B.

Brian HendersonST 371 Homework 0Lecture of 03FEB2011Event Independence Vs. Mutually exclusiveIf events A and B are mutually exclusive of each other then the events will never be independent of each other.0Pr(𝐴 𝐵) 0 Pr(𝐴) 0 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒𝑦 𝑤𝑖𝑙𝑙 𝑛𝑒𝑣𝑒𝑟 𝑏𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟.Pr(𝐴 𝐵) Pr(𝐵)Pr (𝐵)If the events are independent of each other then the Venn diagram can be simplified.Pr(𝐴) Pr(𝐴′ ) 1Pr(𝐵) Pr(𝐴) Pr(𝐴 𝐵)Pr(𝐴 𝐵) Pr(𝐴′ 𝐵) Pr (𝐵)

Homework-0Lecture Notes: ST-371-002(02/03/2011)I.John HarrisonST-371-00202/10/2011Review from Tues. Feb. 1st.a) Conditional Probability: What is the probability event A will happen, given that event Balready happened.𝑃𝑟(𝐴 𝐵) II.Pr (𝐴 𝐵)Pr (𝐵)New Material.a) Definition: Two events, A and B, are Independent if Pr(𝐴 𝐵) Pr (𝐴), and aredependent otherwise.i.Example 1: Given a deck of cards, event B Red Card and event A1 DiamondCard. Are these two events independent?· Pr(𝐴1 ) 0.25· Pr(𝐴1 𝐵) 0.50ANSWER: The two events are dependent.ii.EXAMPLE 2: In addition to the above statements, event A2 Jack Card. Areevents A2 and B independent?· Pr(𝐴2 𝐵) 1113· Pr(𝐴2 ) 13ANSWER: The two events are independent.Note: A2 and B’ are also independent.

John HarrisonST-371-00202/10/2011b) If two events, A and B, are mutually exclusive and A 0 and B 0, are A and Bindependent?· 𝑃𝑟(𝐴 𝐵) Pr (𝐴 𝐵)Pr (𝐵) 0Pr (𝐵)· Pr(𝐴) 0ANSWER: The two events are always dependent.c) If two events, A and B, are independent then the probability tree can be redrawn asshown below:· Pr(𝐴 𝐵) Pr(𝐵) Pr(𝐴 𝐵) Pr(𝐵) Pr (𝐴)· Also, Pr(𝐴 𝐵) Pr(𝐵 𝐴′ ) Pr (𝐵)d) The Following is an example of transferring data from a table to a tree:

Tommy QuinnST 371HW0LECTURE 2/3/11 SUMMARYThis lecture first dealt with conditional probability. Conditionalprobability is the probability of an event given that another eventalready occurred.Ex) Probability of A given BPr(A B) Pr(𝐴 𝐵)Pr (𝐵)Next we talked about Independent events. Two events are independentif P(A B) P(A)Ex) Probability that card drawn in event A is a Jack given event B wasthe drawing of a red card.Pr(A B) Pr(𝐴 𝐵)Pr (𝐵) 25226/52 1/13It was stated that if A and B are mutually exclusive - (A B) 0 then Aand B are never independent.Various examples were then given to demonstrate independent eventson a tree diagram.The next item in the lecture was the gumdrop example. Every person inthe class randomly picked a red or a green gumdrop from a bowl. It wasfound that out of the 13 females in the class, 5 chose red, 8 chosegreen. Out of the 48 males in the class, 24 chose red and 24 chose

green. The below tree diagram was made to compute the probabilitiesof each choice.

B Red Card Pr(A1) .25 (Probability the card is a diamond) A(1) Card is a Diamond Pr(A1 B) .50 (Probability that the card is a Diamond given it is Red) A(2) Card is a Jack Pr(A2

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