Probability (part 3) More on probability.
Probability (part 3)
- 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
Have something that's not a question about this content?
This discussion area is not meant for answering homework questions.
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Thank the author
This is great, I finally understand quadratic functions!
Have something that's not a tip or thanks about this content?
This discussion area is not meant for answering homework questions.
At 2:33, Sal said "single bonds" but meant "covalent bonds."
For general discussions about Khan Academy, visit our Reddit discussion page.
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or thanks in Questions
- a question in Tips & Thanks
- an answer that should be its own question