Probability (part 3) More on probability.
Probability (part 3)
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- Let's keep doing some problems.
- So in the last time I said, what are the chances of not
- getting any heads if I flip the coin-- I was going to say dice,
- but I realized we're dealing with a coin.
- If we flip the coin seven times, if you said, well,
- that's the same thing as getting 7 tails in a
- row, and that's 1/2 to the seventh power.
- Because each trial there's a 1/2 chance of getting the exact
- thing that we want, which in this case is tails.
- And you multiply it times itself seven times
- and you get 1/128.
- It actually turns out that there are actually 1/128
- possible results that we can get, equally probable results.
- And why do I say that?
- Well, what's the probability of getting-- I don't know-- all
- heads and then the last flip I get is tails?
- So heads, heads, heads, heads-- so I'm doing it
- seven times, rights?
- So 1, 2, 3, 4, 5, 6, and then the last time I get a tails.
- If you can think of it, this is exactly 1 particular result out
- of the total results that we can get.
- And what's the probably of this?
- Well, there's a 1/2 chance of getting a heads, 1/2 chance of
- a heads, 1/2 chance of heads-- let me switch colors-- 1/2
- chance of a heads, 1/2 chance of a heads, 1/2 chance of a
- heads, and then 1/2 chance of a tails.
- Once again, this is just 1/2 times 1/2 times 1/2 times
- 1/2 times 1/2 times 1/2, which is equal to 1/128.
- So any particular-- I don't want to say combination because
- we'll learn a little bit about permutations and combinations
- in future videos.
- But any particular set of circumstances we want is 1
- out of 128 of the total number of outcomes.
- That what I should use.
- Instead of circumstances, outcomes.
- If I have any particular outcome-- so this is a
- particular outcome, the way to view it is there are 128
- particular outcomes, so if I choose one of them my odds of
- getting it is 1 out of 128.
- So let me ask you a question.
- What's the probability of getting exactly 1 heads?
- So what does this mean?
- This means I could get-- oh, I could get tails, tails, tails
- and-- let's say we're doing it five times just so I don't
- waste too much time.
- Out of 5 rolls, out of 5 flips.
- So that could be tails, tails, heads.
- That could be-- as you can tell, I switched
- colors arbitrarily.
- That could be tails, tails, tails, head, tails.
- I think you get the pattern.
- Tails, tails, heads, tails, tails.
- That could be tails, heads, tails, tails, tails.
- That could be tail-- oh sorry.
- That could be heads, tails, tails, tails.
- So if we look at this way there's this probability.
- There's actually how many events, how many outcomes,
- would satisfy this statement, exactly 1 heads?
- Well, there's 1, 2, 3, 4, 5.
- And the way you could think of it is we're going to do
- 5 flips, exactly 1 of the flips is going to be heads.
- It could be any 1 of those 5 flips.
- There are 5 situations in which that happens.
- And notice, I said exactly one heads.
- Because if I said at least 1 heads then we would have to
- take into circumstances where we have 2 heads and then it
- becomes more complicated.
- We'll learn more about that when we do combinations.
- So what are the chances of any one of these possible outcomes?
- Well, if we just look at this one.
- There's a chance of 1/2, 1/2, 1/2, 1/2, 1/2, so it's
- 1/2 to the fifth power.
- Which is equal to 1/32.
- And you use the same logic each to these.
- This is 1/32, 1/32, 1/32, 1 out of 32.
- And in general, when I'm flipping a fair coin five times
- in a row there are going to be 32 possible outcomes.
- And each of these is just 1 of the particular outcome.
- And we're essentially saying, out of the 32 outcomes, 5
- satisfy the event that we're looking for-- exactly 1
- heads out of 5 flips.
- So there are 5 that satisfy it out of a total of 32 outcomes.
- And so using that definition only if we can say that all
- of the outcomes are equally probable.
- The probability is 5/32.
- And then you could say, well, let me make it a little
- bit more complicated.
- What is the probability of not getting exactly one head?
- So here there is a lot more circumstances that satisfy it.
- So for example, let me throw out a couple
- that would satisfy it.
- Well, you could get all tails.
- But also, if you got 2 heads that would satisfy it because
- you didn't get exactly 1 head, So this would also satisfy it.
- And so you might say, well, if we do it like this
- it's really complicated.
- Or you could say, well, these 5/32 are the only outcomes that
- don't satisfy-- that do not satisfy this condition.
- So the rest of the what?
- 32 minus 5 is what?
- The other 27 outcomes will satisfy this.
- So you could say that this is equal to-- and this is
- a common trick in a lot of probability problems.
- This is equal to 1 minus the probability of getting exactly
- one heads out of 5 flips.
- And that is equal to-- 1 is the same thing as 32/32.
- Minus-- what's the probability of this?
- We figured it out.
- It's 5/32.
- And that equals 27/32.
- So that's just always something to keep in mind.
- Sometimes when you get a probability problem it seems
- difficult to solve that problem, but it's actually
- not that difficult to solve the opposite problem.
- So you say, well, it's hard for me to directly figure out the
- probability of not getting exactly one heads; I'd have to
- know combinations and all of this.
- But this is the opposite of getting exactly one heads,
- and that's an easier thing for me to figure out.
- So if I can figure out that probability, all of the other
- outcomes would satisfy the opposite one.
- Hope I didn't confuse you.
- So anyway, everything we've done so far, it's been dealing
- with coins and fair coins and that's interesting to a certain
- degree, but let's make it so that the different outcomes
- have different probabilities.
- Let's do free throw percentages because I think this is
- something that we all-- if we watch basketball or play
- any basketball, we're familiar with the idea.
- So let's say that I, and this is not true, I only could
- wish-- let's say that I have an 80% free throw percentage.
- So that means every time I go to take a free throw there's an
- 80% chance there's a basket and that there's a 20% chance
- there's no basket.
- How does this come up with?
- Well, we'll do more on statistics later and they're
- really-- probability and statistics are opposite
- sides of the same I guess coin, you could say.
- But someone says OK, out of the last hundred times Sal took
- free throws he's made 80.
- So in general, he has an 80% chance of making a particular
- free throw, but it might not have been a hundred.
- I might have taken a thousand free throws and I made 800.
- But that's where you come up with the 80% from.
- So that's fair enough.
- So the 80% is p, and then a common notation is
- the 20% is 1 minus p.
- If I have an 80% chance of making the shot, I have a 20%
- chance of missing the shot.
- So my question to you, let's do kind of the same examples
- we did with the coins.
- What's the probability of me making-- I don't know-- 3 shots
- in a row, 3 baskets in a row, or 3 free throws in a row?
- I'm going to say free throws because this is my free
- throw percentage, not my overall shot percentage.
- Well, by the same logic it's going to be equal.
- The first one I would have to get right, then the second one
- I would have to get-- I would have to get a basket and
- there's a 20% I have no basket.
- And then the third one I would have an 80% chance
- of getting a basket.
- So this is-- I didn't draw the rest of the tree.
- It gets pretty broad.
- But in general, just from the coin example you can just
- multiply these probabilities.
- So 80% is the same thing as 0.8, so you get 0.8
- to the third power.
- I don't know, what is that?
- Let me see.
- It is 0.8 times 0.8 times 0.8 is equal to 51.2
- or 0.512-- 51.2%.
- So I have slightly better than even odds of making 3 free
- throws in a row, which is pretty good.
- And as you can see, I don't have an exponent on this
- cheapo Windows provided calculator I use.
- I have to apologize for that.
- So we might to limit our exponent size.
- But anyway, that should hopefully-- well, that's
- a little bit of a taste.
- In the next video I'll do a lot more examples with the free
- throws and the basket, and we'll even learn a little bit
- about conditional probability.
- See you soon.
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