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:6:48

Thomas's favorite color Thomas's favorite colors are blue and green he is one blue shirt one green shirt one blue hat one green scarf one blue pair of pants and one green pair of pants thomas selects one of these garments at random let a be the event that he selects a blue garment and let B be the event that he chooses a shirt which of the following statements are true and they all let's see it before I even read them they all about a deal with probability of event a probability of event B probability of B given a probability of a given B probability of a and B so actually let's just calculate these things ahead of time before we even look at these right over here so let's just think about let's just think about probability of a the probability of a that's the probability that he picks a blue that he selects a blue garment so how many equally likely outcomes are there well there's one two three four five six equally likely outcomes and how many involve selecting a blue garment well there's one two three of the equally likely outcomes involve selecting a blue garment so he has a 3/6 or 1/2 probability of selecting a blue garment now let's what's probability of B what's probability of B and I'll do this in a new neutral color since we're just saying that he's B is just the event that he chooses a shirt so once again there's six possible items equally likely outcomes here and which involve a shirt well there's one there is two so it looks like two of the six involve picking a shirt or we could say the probability of B is equal to one-third now what's the probability of a given B let's write that down what's the probability of we just in a new color what's the probability probability of a given B I'll do those in the colors a given that B has happened so this is saying what's the probability what's the probability probability of a given B is the probability that he picks a blue garment given that he has picked a shirt so this the given B that restricts our outcomes to these two and so the probability that he's picked a blue item well that's one out of the two equally likely ones so there is a 1/2 probability that he picks a that he picks a blue garment given that he's picked a shirt and that's because there's one blue shirt and one green shirt now let's look at the probability of B given a probability of B given a B given probability of B given a so assuming that we've picked a blue garment so assuming we've picked a blue garment so it's either that one that one or that one what's the probability that we have also chose 2 what's the probability that we have also chosen a shirt well there's one two three possibilities equally likely possibilities where we have a blue garment and only one of those involves a shirt so probability of B given a is one-third and then finally we could think about probability of a and B so the probability probability of a and and B a and B so this is the probability of picking a blue shirt so only one out of the six equally likely outcomes is a blue shirt so this one right over here is going to be one one over six so now that we figured out all of that let's see if we can answer these questions the probability of a given B equals the probability of a and that does work out probably if a given B is one-half and that's the same thing as the probability of a the probably that so that Thomas selects a blue government given that he has chosen a shirt is equal to the probability that Thomas selects a blue garment yep that's so they I guess the words are just rephrasing what they wrote here and I guess more mathy notation so this is absolutely true the probability of B given a is equal to the probability of B yep probability of B given a is 1/3 and the probability of B is 1/3 the probability that Thomas selects a shirt given that he's chosen a blue garment is equal to the probability that Thomas selects a shirt yep that's right events a and B are independent events independent events so a two events are independent if the if if let me write it more in math notation these are independent if the probability of a given B is equal to the probability of a then we could say a and B are independent because the probability the probability of a if this is true that this means the probability of a given B actually isn't dependent on whether B happened or not it's the same thing as a probability of a this would lead to these events being independent also if you had probability of B given a is equal to the probability of B same argument that would mean they're independent or if we say that the probability of a and B and B is equal to the probability of a time's the probability of B then these would also mean they're independent and we know that this one's true and the probability of a and B is 1/6 and probability of a times probability of B is 1/2 times 1/3 which is 1/6 so all of these are clearly true so we can say that a and B are independent the probability of a is independent of whether B has happen or not the probability of B happening is independent of whether a has happened or not the outcome event of events a and B are dependent on each other no that's the opposite of saying that they're independent so we're not going to we can cross that out probability of a and B is equal to probability of a times probability of B yet we already said that to be true 1/6 is 1/2 times 1/3 the probability that Tom selects a blue garment that is a shirt is equal to the probability that Tom selects a blue garment multiplied by the probability that he selects a shirt yep that's absolutely absolutely right so actually this was a lot of these statements are true the only one that's not is that the outcome of events a and B are dependent on each other