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.