This video is provided as supplementary

material for courses taught at Howard Community

College and in this video I want to talk about

probability for independent and dependent events. So here’s our example. I’ve got some pieces that I took

from a Scrabble game. There are two squares and they have A’s on them. And there are two squares, two pieces

and they have E’s on them. If I put these pieces into a bag and

shook the bag, and just randomly drew one of the pieces, I

would have a 50 percent chance of getting an A, since there are two A’s and four pieces all

together. I’d also have a 50 percent chance getting an E because there are two E’s and four pieces all

together. So I could represent these probabilities with a tree. I’d have these two branches

for the tree. One branch would represent getting an A

and the other branch would represent getting an E.

And the probability for getting an A would be 1/2. The probability for

getting an E, would be 1/2. Now if I want to play this

game again, or run this experiment again,

I’ve got two choices. I could either take the letter that I picked

and put it back in the bag or I could not put it back in the bag. We’ll get a different probability the second

time around depending on whether we replace that

letter or not. So let’s look at both of those options. If I put the letter back in the bag,

then once again there are two A’s in the bag

and two E’s in the bag. So the probabilities is second time

around haven’t changed at all.

They’re independent of what I picked the first time. If I want

the probability of getting an A after I got an A the first time, for the second time I’m still gonna have

a 1/2 probability of getting an A, and I’ll have a 1/2 probability of getting an E. And the same kind of

probabilities would happen if I picked an E first time around.

If I put it back in the bag, if I replaced it, then the probability for

getting an A on the second turn would be 1/2 and the probability of getting an E

on the second turn would be 1/2. Now we can take this tree and figure out

what the probability would be of getting two A’s in a row. All we have to

do is follow the branch of the tree that

that gives me two A’s in a row and multiply the probabilities each time. So I’ve got a probability of 1/2 from

the first time that I got an A and a probability of 1/2 from

the second time that I got an A. So that’s 1/2 times 1/2 and that multiplies out to 1/4. Now when I did that I replaced the piece that I took. Whatever I took, I replaced it, I put it back in the bag.

So we could call that probability with replacement. We can also talk about that as being two independent events, because the probabilities for the second time

around did not depend on what I picked the

first time around. The second time around my probability of

getting an A was 1/2 and my probability

of getting an E was 1/2. And that didn’t depend on whether

I got an the first time or an E the first time. So these are called independent events. In other words, the probability for the second event was independent of the probability for the

first event. Now let’s see what would have happened

if we played that game differently. Let’s say that we did this with what are called ‘dependent events’.

So we would start out the same way. We would have the same four letters. I would put them in the bag

and then randomly pick one of the letters. So my probability once

again for getting an A is 1/2 and my probability for getting

an E is 1/2. Now let’s say that this time I don’t replace the letter that I picked.

So let’s say that I get an A and I don’t put it back in the bag. So now on the second turn, what’s my

probability for getting an A? Well it’s no longer 1/2. I’ve only

got one A left in the bag and there are three letters in the bag.

So my probability for getting an A the second time around is only 1/3. My probability for

getting an E is 2/3. If I had gotten an E the first time and didn’t

replace it, then my probabilities would

change again. If I had gotten an the first time and didn’t

replace it, there would be two A’s in the bag and only E. So now my probability

of getting an A would be 2/3 and my probability of getting an E would be 1/3. And if I want to know the probability of

getting two A’s in a row, I” figure that out the same way I did

before. I’ll take the branch of the tree that has two A’s in a row and I’ll multiply 1/2, the

probability from the first time, times 1/3, the probability from the

second time. 1/2 times 1/3 is 1/6. So these are called ‘dependent events’, because the

probability that I end up with the second time around depend on what

happened the first time The probability the second time for

getting an A or getting an E are different if I got an A the first time

than if I got an E the first time. And that happened because I took the pieces out and did not replace them.

So this is probability without replacement. So just to review this one was time quickly… The probability for independent events

doesn’t change from one event to the next depending on

what happened previously. If we’re dealing with pieces like

Scrabble pieces or cards or something like that, it would mean that we replace the pieces

that we took the first time, so the odds, so the probabilities remain the same.

If we’re dealing with dependent events, then the probabilities for the second event are going to depend on what happens on

the first event. And in a game like this one it’s because

we played the game without replacement. So that’s basically independent and dependent events and their probabilities. Take care, I’ll see you next time.