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## Conditional probability and independence

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# Analyzing event probability for independence

## Video transcript

Voiceover:Tomas's favorite colors, Tomas's favorite colors
are blue and green. He has one blue shirt, one green shirt, one blue hat, one green scarf, one blue pair of pants, and one green pair of pants. Tomas selects one of
these garments at random. Let A be the event that
he selects a blue garment. Let B be the event that
he chooses a shirt. Which of the following
statements are true? And they all, let's see, and before I even read them, they all deal with the
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, 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. So what's the probability of B? What's the probability of B? And I'll use neutral colors since we're just saying he's B 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's two, so it looks like two of the six involve picking a shirt. Or we could say the probability
of B is equal to 1/3. Now what's the probability of A given B? Let's write that down. What's the probability of let's use a new color, What's the 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 of A given B? The 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. So the probability that
he picks a blue item well, that's one out of the
two equally likely ones. So that is a 1/2 probability that he picks a blue garment given that he's picked a shirt, and that's because there is one blue shirt and one green shirt. Let's look at the
probability of B given A. Probability of B given A. B given, probability of B given A. So assuming we have picked a blue garment. So assuming we've picked a blue garment. So it's either that one or that one. What's the probability that we have also, What's the probability that
we have also chosen a shirt? Well, there's one, two,
three possibilities, equally likely possibilities that we have a blue garment. And only one of those involve a shirt. So the probability of B given A is 1/3. And then finally we can think about probability of A and B. So the probability, probability, of A 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've 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. Probability of A given B is 1/2. And that's the same thing
as the probability of A. The probability that
Tomas likes a blue garment given that he has chosen a shirt is equal to the probability that Tomas likes a blue garment. Yep, that's exactly. So I guess the words are just rephrasing what they wrote here in
a more mathy notation. So this is absolutely true. The probability of B given A is equal to the probability of B. Yep, the probability of B given A is 1/3. The probability of B is 1/3. The probability that Tomas selects a shirt given that he has chosen a blue garment is equal to the probability
that Tomas selects a shirt. Yep, that's right. Events A and B are independent events. Independent events. So two events are independent if, well let me write it in math notation. These are independent if the probability of A given B is equal to the probability of A. Then we can say A and B are independent. Because the probability of A, then if this is true then this means the probability of A given B isn't dependent on
whether B happened or not. It's the same thing as
the probability of A. This would lead to these
events being indpendent. So if you had the probability of B given A is equal to the probability of B. Same argument. That would mean they are independent. Or, if we said that the
probability of A and B is equal to the probability of A times the probability of B then this also means they are independent. We know that this one is true. The probability of A and B is 1/6. The probability of A
times the 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 happened or not. The probability of B happening is independent of whether
A has happened or not. The outcome of events A and B are dependent on each other. No. That's the opposite of
saying they are independent. So we can cross that out. Probably of A and B is equal to the probability of A
times the probability of B 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 is absolutely right. So actually this is, a lot
of these statements are true. The only one that is not is that the outcome of events A and B are dependent on each other.