Conditional Probability

1 Conditional ProbabilitySometimes our computation of the Probability of an eventis changed by the knowledge that a related event hasoccurred (or is guaranteed to occur) or by some additionalconditions imposed on the example, based on a .292 batting average for 2016, wemight assign Probability 29% to Kris Bryant having a hit inhis first at-bat of suppose (closer to opening day), we learn that thepitcher Bryant will be facing in his first at-bat will beleft-handed. We might want to (indeed we should) use thisnew information to re-assign the ProbabilityBased on the data that Bryant had a .314 batting averageagainst left-handed pitching in 2016, we might now assignprobability 31% to him having a hit in his first at-bat new Probability is referred to as aconditionalprobability, because we have someprior informationabout conditions under which the experiment will information may change thesample spaceand thesuccessful event ProbabilityExampleLet us consider the following experiment.

2 Acard is drawn at random from a standard deck of that there are 13 hearts, 13 diamonds, 13 spades and13 clubs in a standard deck of H be the event that a heart is drawn,Ilet R be the event that a red card is drawn andIlet F be the event that a face card is drawn, where theface cards are the kings, queens and jacks.(a) If I draw a card at random from the deck of 52, what isP(H)?1352= 25%. Conditional Probability (b) If I draw a card at random, and without showing youthe card, I tell you that the card is red, then what are thechances that it is a heart?1326= 50%. Because you knowthat the card drawn is red, the Sample Space of all possibleoutcomes has size 26 (there are 26 red cards), not 52.

3 Ofthe 26 red cards, 13 are hearts, so there are 13 we are calculating the Probability that the card is aheart given that the card is red. This is denoted byP(H|R), where the vertical line is read as given . Noticehow the Probability changes with the prior also that we can think of the prior information asrestricting the sample space for the experiment in this can think of all red cards or the setRas areducedsample Probability (c) If I draw a card at random from the deck of 52, what isP(F)?There are 12 face cards soP(F) =1252=313 (d) If I draw a card at random, and without showing youthe card, I tell you that the card is red, then what are thechances that it is a face card ( what isP(F R))?

4 There are 6 red face cards and 26 red cards soP(F R)=626= that in this case, theprobability does not change even though both the samplespace and the event space do ProbabilityRecap: the Probability that the card is a heart given (theprior information) that the card is red is denoted byP(H R)Note thatP(H R)=n(H R)n(R)=n(H R)/n(S)n(R)/n(S)=P(H R)P(R).This Probability is called theconditional Probability ofH given ProbabilityDefinition: IfAandBare events in a sample spaceS,withP(B)6= 0, theconditional probabilitythat anevent A will occur, given that the eventBhas occurred isgiven byP(A B)=P(A B)P(B).If the outcomes ofSare equally likely, thenP(A B)=n(A B)n(B).

5 From our example above, we saw that sometimesP(A B)=P(A) and sometimesP(A B)6=P(A). WhenP(A|B) =P(A), we say that the eventsAandBareindependent. We will discuss this in more detail in the Conditional ProbabilitiesExampleConsider the data, in the following table,recorded over a month with 30 days:WeatherMoodSNSG96NG114On each day I recorded,whether it was sunny, (S), ornot, (NS), and whether mymood was good, G, or not(NG).(a) If I pick a day at random from the 30 days on record,what is the Probability that I was in a good mood on thatday,P(G)? The sample space is the 30 days underdiscussion. I was in a good mood on 9 + 6 = 15 of them soP(G) =1530= 50%.

6 Calculating Conditional probabilities (b) What is the Probability that the day chosen was aSunny day,P(S)?The sample space is still the 30 days under discussion. Itwas sunny on 9 + 1 = 10 of them soP(S) =1030 33%.(c) What isP(G S)?P(G S)=P(G S)P(S). Hence weneed to calculateP(G S). Here the sample space is stillthe 30 days:G Sconsists of sunny days in which I am ina good mood and there were 9 of them. HenceP(G S) =930. ThereforeP(G S)=9301030=910= 90%.Calculating Conditional probabilities (d) What isP(S G)?P(S G)=P(G S)P(G)=9301530=915=35= 60%.In this example,P(G S)6=P(S G).Note that if we discover thatP(A B)6=P(A), it does notnecessarily imply a cause-and-effect relationship.

7 In theexample above, the weather might have an effect on mymood, however it is unlikely that my mood would have anyeffect on the Conditional ProbabilitiesExampleOf the students at a certain college, 50%regularly attend the football games, 30% are first-yearstudents and 40% are upper-class students ( , non-firstyears) who do not regularly attend football games.(a) What is the Probability that a student selected atrandom is both is a first-year student and regularly attendsfootball games?We could use an algebra approach, or a Venn diagramapproach. We ll do the latter. LetRbe the set of studentswho regularly attend football games; letUbe the set ofupper-class students, and letFbe the set of first-yearstudents.

8 Note in this example thatUdoes not stand forthe UNIVERSAL SET ( if you want a relevant universal setit isF U). We are givenP(R) = 50%;P(F) = 30%;P(U R ) = 40%.Calculating Conditional ProbabilitiesWe knowU Fiseverybody,U F= andR U we can fillin four 0 s as Conditional ProbabilitiesNext identify what you are yellow region isU R and we knowP(U R ) = 40%.Calculating Conditional ProbabilitiesWe also know the val-ues for the diskRandthe Conditional ProbabilitiesSinceF Uiseverybody,P(F U) = the Inclusion-Exclusion Principlewe see we can workout the unknown Conditional ProbabilitiesCalculating Conditional ProbabilitiesSinceP(R) = 50 wecan fill in the last Conditional ProbabilitiesSinceP(F) = 30 wecan fill in the lastbit wecan give the answerto part a):P(F R) =.

9 2 Calculating Conditional probabilities (b) What is the Conditional Probability that the personchosen attends football games, given that he/she is a firstyear student?P(R F)=P(R F)P(F)= 67%.Calculating Conditional probabilities (c) What is the Conditional Probability that the person is afirst year student given that he/she regularly attendsfootball games?P(F R)=P(R F)P(R)= Conditional ProbabilitiesExampleIfSis a sample space, andEandFare eventswithP(E) =.5,P(F) =.4 andP(E F) =.3,(a) What isP(E F)?P(E F)=P(E F)P(F)= 75%.(b) What isP(F E)?P(F E)=P(E F)P(E)= 60%.A formula forP(E F).We can rearrange the equationP(E F)=P(E F)P(F)to getP(F)P(E F)=P(E F).

10 Also we haveP(F E)=P(E F)P(E).orP(E)P(F E)=P(E F).This formula gives us amultiplicative formulaforP(E F). In addition to giving a formula for calculatingthe Probability of two events occurring simultaneously, it isvery useful in calculating probabilities (E F) =P(E)P(F E).The Probability thatEand thenFwill occur is theprobability ofEtimes the Probability thatFhappensgiventhatEhas : IfP(E F)=.2 andP(F) =.3, findP(E F).P(E) =P(E F) P(F) = = = 6%.Example: The Probability that it will be 30 F or belowtomorrow morning is When the temperature is thatlow, the Probability that my car will not start is Whatis the Probability that tomorrow morning it will be 30 F orbelowandmy car will not start?