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

Current time:0:00Total duration:8:51

what I hope to do in this video is give ourselves a more intuitive sense of probabilities so let's go back to an example that we've seen before we're rolling a fair six-sided die and so there are six possibilities we could get a 1 a 2 a 3 a 4 a 5 or a 6 now let's say we ask ourselves what is the probability of rolling of rolling a number that is less than or equal to less than or equal to 2 what is this going to be well there are six equally likely possibilities in rolling less than or equal to 2 well that means I'm either rolling a 1 or a 2 so 2 1 2 of the six equally likely possibilities meet my constraints so there is a 2/6 probability of rolling a number less than or equal to 2 or I could just rewrite that as an equivalent fraction I could say there's a 1/3 probability I can go either way now let's ask ourselves another question what is the probability of rolling a number greater than or equal greater than or equal to 3 well once again there are six equally likely possibilities and how many of them involve rolling greater than or equal to three let's see one two three four these possibilities right over here the roll of three or four five or six so four out of the six equally likely possibilities or I could rewrite this as an equivalent fraction as 2/3 so what's more likely rolling a number that's less than or equal to 2 or rolling a number that's greater than or equal to 3 well you can see right over here the probability of rolling greater than equal to 3 is 2/3 while the probability of rolling less than or equal to 2 is only 1/3 this number is a greater so this has a greater probability or another way to thinking about it rolling greater than or equal to 3 is more likely than rolling less than or equal to 2 in fact not only is it more likely you see that 2/3 is twice 1/3 this right over here is twice as likely you're twice as likely to roll a number greater than or equal to three then you are to roll a number less than or equal to two and you could even see right over here you have twice as many possibilities of the six equally likely ones four versus two four versus two here and so you say okay I get it Sal you know if the probability is a larger number the event is more likely it makes sense and in this case it's twice that the number is twice as large so it's twice as likely but what's the what's the range of possible probabilities how low can a probability get and how high can a probability get so let's let's think about the first question so how low how low can a probability go how low so what's the lowest probability that you can imagine for anything well let's give ourselves let's give ourselves a little bit of a experiment let's ask ourselves the probability of rolling of rolling a seven well once and and pause the video and try to figure it out on your own well there are six equally likely possibilities and how many of them involve rolling a seven well none of them it's impossible to roll a seven so none of the six so we could say this probability is zero and if you see a probability of zero someone says the probability of that thing happening is zero that means it's impossible that means and no in no world can that happen if it's exactly zero so this right meet here the probability is zero that means it is impossible it it is impossible now how high can a probability get so how how high can a probability get well let's think about let's say probability of rolling rolling any number any number from one to six well have six equally I have six I have six equally likely possibilities and any one of those six meets this constraint I would have rolled a number any number from one to six including one and six so there's equally likely possibilities and so the probability is one and so if someone says the probability is zero it's impossible and if someone says the probability is 1 that means it's definitely going to happen going going going it's definitely going to happen so the maximum probability for anything is 1 the minimum probability is 0 you don't have negative probabilities and you don't have probabilities greater than 1 and then you might be thinking wait wait you know I've seen things that you know they look like larger numbers than 1 and you're probably thinking of seeing this as a percentage so 1 as a percentage you could also write this as 100% this right over here as a percentage is 100% but 100% is the same thing as 1 you can't have a probability at 110 percent 110 percent would be the same thing as 1.1 now this is really interesting because you would often see someone say hey you know something for sure is going to happen or something is impossible but even a lot of the things that we think for sure going to happen there's some probability or some chance that they don't happen so for example you might hear someone say well what's the probability that the Sun will rise tomorrow well you might say it's going to happen for sure but you got to remember you know some type of weird cosmological event might occur some kind of strange huge planet-sized object in space might come and knock the earth out of its rotation who knows what could happen all of these have a very low a very low likelihood very very very very very very low likelihood but it's hard to say it's exactly what if I said the probability that the Sun will rise Sun will rise tomorrow tomorrow instead of saying 1 I would probably say it's 0.999 and I would I would throw a lot I would throw a lot of nines over here I wouldn't say it's 0.9 repeating forever and actually there's an interesting proof stat 0.9 repeating forever is actually the same thing one which is a little counterintuitive but I would say there's a very high probability but even something is such a high probability it's going to be close to one but I won't say it's exactly the one because there could be some kind of quasar that blasts us with gamma rays or who knows what might happen but it's a very very high probability same same thing you know the probability the probability here probability that you know my my my my pet my pet my pet gopher my pet gopher could compose could come could write the next great novel writes writes a novel and actually not just a novel a great novel just a novel would be that impressive for Grover but writes a great novel well once again you know this gopher sitting there typing at a keyboard it would seem somewhat random but there is some probability that it actually does it there's some chance it does it so I would put this at a very low and I would say it's exactly zero if we had a an infinite number of Gophers doing this forever who knows maybe one of them might write that great novel in fact if we had an infinite what number doing it forever eventually a lot of people would say it at some point you would but it just one go over trying to write a novel what's the probability they write a great novel well I would say it's pretty close to zero I throw a lot of zeroes I would throw a lot of zeroes here and at some point you might have something like this so once again not absolutely impossible but pretty close to pretty pretty close to impossible and so big takeaways higher probability more likely the lowest probability you can get to zero highest probability is one if your probability is more if you know when you talk about coin flipping if you say the probability of heads for a fair coin and you say well that's one-half that means it's equally likely to happen or not happen so anything that has a larger probability than 1/2 it's more likely to happen than not anything that has a probability less than 1/2 it's less likely to happen than not