If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Calculating conditional probability

## Video transcript

rahul 'he's two favorite foods are bagels and pizza let a represent the event that he eats a bagel for breakfast and let B represent the event that he eats pizza for lunch fair enough on a randomly selected day the probability that Rahul will eat a bagel for breakfast probability of a is 0.6 let me write that down so the probability that he eights eats a bagel for breakfast is 0.6 the probability that he'll eat a pizza for lunch probability of event B so the probability of we did that in that red color the probability of event B the eats a pizza for lunch is 0.5 and the conditional probability that eats a bagel for breakfast given that he eats a pizza for lunch so probability of event a happening that he eats a bagel for breakfast given that he's had a pizza for lunch is equal to 0.7 which is interesting so let me write this down the probability of a given given that B is true given B is not 0.6 it's equal to 0.7 and because these two things are not the same because probability of a by itself is different than the probability of a given that B is true this tells us that these two events are not independent that we're dealing with the dependent probability this shows us the fact that B being true has changed the probability of a being true this tells us that a and B are dependent d pendant and so when we start thinking we'll actually let's just said let's just before I start going on on my little soapbox about dependent events let's just think about what they actually want us to figure out so the probability the probability that he eats the probability of a given B is equal to 0.7 that's what we wrote right over here based on this information what is the probability of B given a so they want us to figure out the probability of be given probability of B given a that's what they wants to figure out the conditional probability that Rahul eats pizza for lunch given that he eats a bagel for breakfast rounded to the nearest hundredths so how do we think about this and I encourage you to pause this video before I work through it so I'm assuming you've given a go at it so the best way to tackle this is just just think about well what's the probability that both a and B are going to happen well the probability of a and B happening and let me do this in a new color the probability of a and B happening a and B I want to stay true to the colors is equal to there's a couple of ways you could you could write this out this is equivalent to this is equivalent to the probability probability of a given B given B times the probability of B and hopefully that makes sense it probably the B happens and that given that B it happened the probability that a happens and that would also be equal to so obviously this is a and B is happening or you could view it the other way around you could view it as the probability that B the probability that B given a happens given a happens times the probability of a times the probability of a this also makes sense what's what what what's the probability that a happened and that given a happen what's the probability of B you multiply those together you get the probability that both happened so why is this helpful for us well we know a lot of this information we know the probability of a given B is 0.7 so let me write that I'll scroll down a little bit this is 0.7 we know that the probability of B is 0.5 so this is 0.5 so we know that the probability of a and B is the product of these two things that's going to be 0.35 7 times 5 is 35 or I guess you could say half of 0.7 is 0.35 0.5 of 0.7 and that is going to be equal to what we need to figure out probability of B given a times probability of a but we know probability of a we know that that is 0.6 we know that this is 0.6 so just like that we could we've set up a situation in equation where we can solve for the probability of B given a the probability of B given a notice let me just rewrite it or right over here actually I'll write this part first the pro point six zero point six times the probability of B given a times that right over there and I'll just copy and paste it so I don't have to keep changing colors that over there is equal to 0.35 is equal to 0.35 and so to solve for the probability of B given a we can just divide both sides by 0.6 0.6 0.6 and we get the probability of B given a is equal to like in our calculator out so 0 0.35 divided by divided by 0.6 and we deserve a little bit of a drumroll here is 0.5 8 3 3 keeps going they tell us to round to the nearest hundredth so it's 0.5 8 is approximately is approximately 0.5 8 so another so this is equal to 0 or I'll say approximately equal to 0.58 once again verifying that these are dependent the probability that B happens given a has is true is higher than just the probability that B by itself or that without knowing anything else just the probability of B is lower than the probability of B given that you know given that you know a has happened or event a is true and we're done