when the values are substituted into the formula here is the result 
p b a 1 6 1 2 1 3 p b a 1 6 1 2 1 3 p b a 1 6 1 2 1 3this means that given the die rolled is even the probability that this number is also greater than four is 1 3 another scenario involves a student applying for admission to a college who hopes to get a scholarship and a stipend for books meals and housing the steps to determine the conditional probability of getting a stipend and the scholarship are as follows step 1 understand the scenariofirst the student wants to know the likelihood of being accepted to the university then if accepted the student would like to receive an academic scholarship moreover if possible the student would also like to receive a stipend for books meals and housing if they get the scholarship step 2 define the eventsthere are three events step 3 calculating the probability of being accepted event a the university accepts 100 out of every 1 000 applicants who have applications similar to the student s thus the probability of a student being accepted is p a 100 1000 0 10 or 10 step 4 determining the probability of receiving a scholarship once accepted p b a it s known that out of the students accepted 10 out of every 500 receive a scholarship thus the probability of receiving a scholarship given acceptance is as follows p b a 10 500 0 02 2 p b a 10 500 0 02 2 p b a 10 500 0 02 2 step 5 calculating the probability of being accepted and receiving a scholarshipto calculate the probability of being accepted and also receiving a scholarship the likelihood of acceptance is multiplied by the conditional probability of receiving a scholarship given acceptance p a b p a p b a 0 1 0 02 0 002 0 2 p a b p a p b a 0 1 0 02 0 002 0 2 p a b p a p b a 0 1 0 02 0 002 0 2 step 6 determining the probability of receiving a stipend having gotten a scholarship p c b it s also known that among the scholarship recipients 50 receive a stipend for books meals and housing thus p c b 0 5 50 step 7 calculating the probability of being accepted receiving a scholarship and receiving a stipendto calculate the probability of a student being accepted receiving a scholarship and then also receiving a stipend the probabilities of the events are multiplied p a b c p a p b a p c b 0 1 0 02 0 5 0 001 0 1 p a b c p a p b a p c b 0 1 0 02 0 5 0 001 0 1 p a b c p a p b a p c b 0 1 0 02 0 5 0 001 0 1 this step by step breakdown illustrates how the probabilities for each scenario are calculated using basic probability formulas and conditional probability conditional probability vs joint probability and marginal probabilitylet s now differentiate calculating conditional probability from other kinds of probability the example this time is a regular deck of cards two events are defined a standard deck has 52 cards divided into four suits hearts diamonds clubs and spades hearts and diamonds are red and clubs and spades are black each suit has 13 cards ace then two through 10 and then the face cards jack queen and king the deck contains 26 red cards 13 hearts and 13 diamonds thus the probability of drawing a red card is p b 26 52 1 2 within the red cards are a four of hearts and a four of diamonds therefore if a red card has to be drawn a subset of the deck that includes only these 26 red cards needs to be considered given that a red card has been drawn the probability of it being a four is calculated as follows p a b n u m b e r o f r e d f o u r s t o t a l n u m b e r o f r e d c a r d s 2 26 1 13 p a b number of red fours total number of red cards 2 26 1 13 p a b numberofredfours totalnumberofredcards 2 26 1 13the marginal probability p a is the probability of an event a happening on its own it does not consider the occurrence of any other event since event a is drawing a four p a is calculated by dividing the number of fours by the total number of cards in the deck p a n u m b e r o f f o u r s i n t h e d e c k t o t a l n u m b e r o f c a r d s i n d e c k 4 52 1 13 p a number of fours in the deck total number of cards in deck 4 52 1 13 p a numberoffoursinthedeck totalnumberofcardsindeck 4 52 1 13joint probability is the likelihood of two or more events happening at the same time this is denoted as p a b the probability of events a and b occurring assuming that the previous events are the same that is event a is the occurrence of drawing a card that is a four and event b is drawing a red card we can find the joint probability of drawing a card that is both a four and red there are two cards that meet both criteria the four of hearts and the four of diamonds thus the joint probability of drawing a card that is both a four and red is calculated as follows p a b n u m b e r o f r e d f o u r s t o t a l n u m b e r o f c a r d s 2 52 1 26 p a b number of red fours total number of cards 2 52 1 26 p a b numberofredfours totalnumberofcards 2 52 1 26bayes theorem and conditional probabilitybayes theorem is used to calculate conditional probabilities when dealing with uncertain events in investing this allows you to update your probability estimates of a market outcome when you get new relevant data for example suppose you wanted to know the probability that the s p 500 would return a positive percentage this year given initial gross domestic product gdp figures in that case you d start with bayes theorem considering the index s historical return rates to get an initial estimate of projected economic expansion you would then revise this first probability using the latest gdp estimates to provide more refined probability assessments incorporating all evidence as the year progresses while a bit complex mathematically bayes theorem is quite logical if an investor discovers new economic information relevant to potential market returns it makes sense to integrate this data to get a more precise calculation the 18th century english minister thomas bayes devised this statistical technique which remains central across financial modeling and other fields requiring predictions under uncertain conditions bayes theorem is well suited to and widely used in machine learning 