Conditional Probability Background Conditional Probability is just a process of simplification When we set a condition we eliminate possibilities or insist that an event has become certain This allows us to reassign the probabilities in a model proportionally Discussion Let s look at some situations and then set some conditions for them Example You roll two dice The concern is for the total of the values on the dice This is sometimes called the point in dice games 1 1 You know the probability of getting a point of 7 is 2 1 calculated easily by counting There are 36 outcomes as shown to the right Notice by creating the lattice then 3 1 putting the total on the edges we can count along diagonals to get the successful results This is a useful technique any 4 1 time we use pairs of numbers 5 1 6 1 The probability of getting a seven is P 7 6 1 36 6 Similarly the probability of getting a point of 5 is P 5 2 3 4 5 6 7 1 2 1 3 1 4 1 5 1 6 8 2 2 2 3 2 4 2 5 2 6 9 3 2 3 3 3 4 3 5 3 6 10 4 2 4 3 4 4 4 5 4 6 11 5 2 5 3 5 4 5 5 5 6 12 6 2 6 3 6 4 6 5 6 6 5 36 6 7 1 5 1 6 8 2 4 2 5 2 6 9 3 3 3 4 3 5 3 6 10 4 2 4 3 4 4 4 5 4 6 11 5 1 5 2 5 3 5 4 5 5 5 6 12 6 1 6 2 6 3 6 4 6 5 6 6 Now let s change the situation Example Suppose you know for a fact that the result of a roll of the two dice is greater than 5 What is the probability that you get a point of 7 Our condition is a total more than 5 Since the point is greater than 5 none of the upper left part of the lattice applies So we are reduced to the 21 possibilities to the right So the probability of getting a point of 7 is P 7given a point greater than 5 6 2 21 7 8 2 6 9 3 5 3 6 10 4 4 4 5 4 6 11 5 3 5 4 5 5 5 6 12 6 3 6 4 6 5 6 6 To simplify the writing we use a notation for conditional probability We let stand for given Here is an example P 7given a point greater than 5 P 7 point 5 6 2 Arizona State University Department of Mathematics and Statistics 1 of 6 Conditional Probability Example What is the probability of getting an even point given that the point is greater than 7 The condition has shrunk the pool of possibilities even more We now have only 15 possibilities We can see that points of 8 10 and 12 remain 5 3 1 9 3 15 15 5 Here is a cute example of how conditional probability works By counting we have this result P even point 7 Example The king has two siblings What is the probability that he has an older sister There doesn t seem to be a condition here but in traditional monarchies the king is the oldest male child Other male children were sent off to crusades hopefully to die honorably in battle or some how be taken care of So the condition implicitly is that given that any possibly combination where there is a male child the oldest is the king We can also discount all possibilities where there are no male children at all Now the sample space for a three child family is bbb bbg bgb bgg gbb gbg ggb ggg where b is boy and g is girl Now since each outcome is equally likely the probability for any outcome is 0 125 When we apply the condition we get a new smaller sample space with seven uniform outcomes bbb bbg bgb bgg gbb gbg ggb Remember each of these remains equally likely Someone may have noted that the only impossible situation is the ggg outcome So we could have eliminated that single outcome To make it easier to see what is good for us older sister we can change the first b to a K Kbb Kbg Kgb Kgg gKb gKg ggK Counting where there is a g before the K we get only gKb gKg ggK So there are three ways the king has an older sister out of the seven Therefore P older sister king 3 7 A Conditional Formula Let s try to think this through using a different more calculation driven process Since we are reapplying the probabilities to the reduced model let s use the idea that each acceptable outcome remains equally likely fraction of success We could ask What is the proportional probability of success out those outcomes remaining after we shrink the sample space The calculation looks is to the right Notice this is the same numerical value This is not coincidence In the next example we will see that it holds true again 2 of 6 Arizona State University Department of Mathematics and Statistics P g older than K P conditional space P gKb or gKg or ggK P condtional space 1 8 18 18 7 8 3 7 Conditional Probability Example Professor Jones uses three test forms when he gives a test He tries to make sure each one is equivalent to the others but sometimes it just doesn t happen that way In a recent test he observed the results to the right out of his 50 students taking the test Because his students work hard and really study and practice no one failed the test Professor Jones is a little concerned The number of students getting A s on the Form A B C Total third form seems high Maybe the third form was easier or the other forms were harder We ll get to that is a moment For now let s just handle a few conditional probability questions 1 2 4 4 10 2 6 8 11 25 3 9 4 2 15 Total 17 16 17 50 Question 1 What is the probability that a student used Form 1 given that a student made an A This is simple We restrict ourselves to the A column So P 1 A 2 0 1176 17 Compare that result to What is the probability that a student made an A That result is P A 17 0 34 50 Question 2 What is the probability that a student made an A given that they used form 1 2 This is also simple We restrict ourselves to the Form 1 row So P A 1 0 20 10 Compare that result to What is the probability that a student used form 1 That result is P 1 10 0 2 50 Note that the condition …