Probability (part 4) More on free throws.
Probability (part 4)
- So we started doing some free-throw percentage
- problems in the last video, so let's continue.
- So the first time I said if I had an 80% free-throw
- percentage, what are my chances of getting three
- free throws in a row?
- And I said, well, it's 51.2% chance.
- Let's do another scenario, and let's just assume that my
- free-throw percentage stays at 80%.
- And let's say that-- I don't know-- let's say my team
- is two points behind.
- I'm making this up on the fly, so we're two points behind.
- No, let's say we're one point behind.
- And I'm up at the free-throw line, and I want to say, what
- is the chance-- and let's say I have three free throws.
- Someone fouled me when I was shooting a 3-pointer.
- So we're one point behind, there's one second left, and
- right when I was going to take the winning free throw-- or
- not the winning free throw.
- The winning 3-point shot, someone fouls me.
- So I get three free throws.
- And once again, I have an 80% free-throw percentage.
- My actual percentage is probably something
- more like 20%.
- But anyway, I have three free throws.
- So my question is, what are our chances of winning?
- What is the probability of my team winning?
- Not tying, winning.
- So in order to win-- well, actually let's
- do the tying first.
- So what's the probability of tying?
- How do you spell-- tying, OK.
- What's the probability of a tie?
- Well, the probability of a tie is going to be equal to the
- probability of me getting at least one free throw.
- You might say OK, that's the probably of getting exactly 1
- times the probability of getting 2 out of 3 plus
- the probability of getting 3 out of 3.
- But like we saw in-- I think it was either the last video or
- the video before-- that is identical equal to the
- probability or, let's see, 1 minus the probability of not
- getting any free throws.
- The probability of getting at least one is essentially the
- probability of me not missing all three.
- Hopefully that makes some sense.
- So let me erase all this stuff that I did up on top, and
- we can to figure out the answer to this problem.
- Let me erase all of this, erase all that.
- Erase that.
- Let me erase this.
- We know I have an 80% probability of making
- any one shot.
- So if we go back to the problem-- so this is my
- free throw percentage.
- So what is the probability of me not making any shots, not
- making any of my free throws?
- Well, my probability of not making one free throw is 20%.
- 20% chance of not making any free throws, or not making
- any-- say I take a shot, I have a 20% of no basket, or not
- making the free throw.
- So if I'm taking three shots, the shot of I miss all three
- is going to be equal to 20% times 20% times 20%.
- And that equals what?
- This 0.2 times 0.2 is 0.04 times 0.2.
- And then let's see.
- What's 0.04 times 0.2?
- Let's see, that'll be an 8 and I have 1, 2, 3 numbers
- by the decimal place.
- It's 1, 2, 3.
- So 0.008.
- So I have a 0.008 chance of missing all of the free throws.
- Or another way you could view this is, is this is equal to--
- a percentage is just this times 100 so there's 0.8% chance of
- missing all of the free throws.
- Or another way of viewing this is I have-- so what's my chance
- of making at least one?
- Well, it's going to be 1 minus this.
- So what's 1 minus 0.-- this is a 0.
- Well it's 99.2%.
- Yeah, that's 99.2%.
- Or you could also view it as 0.992.
- You take 1 minus this, you get 0.992, which is the
- same thing as 99.2% chance.
- So I have a 99.2% chance of at least tying the game.
- So it's pretty high if you have someone at the free-throw line
- with a 80% free-throw percentage and they
- have three shots.
- Now, what if I only had two shots?
- Let's say I got fouled when I was thinking a 2-pointer
- and I only have two shots.
- Well, in that case, in order to tie the game I have to get at
- least one, but I only have two shots.
- So it would be 1 minus the chances of me
- missing both shots.
- What's the chances of missing two shots?
- Well, the chance of missing two shots in a row-- it's 4%.
- 20% times 20%.
- It's a 4% chance of missing two shots, two in a row.
- So my probability of getting at least one, assuming that I am
- taking two shots, is going to be 1 minus this.
- I have a 96% probability of at least one shot if I take two.
- So that's also pretty high.
- And of course, what's my probability if I
- only have one shot?
- Well, it's 80%.
- So hopefully that gives you a little bit of framework next
- time you watch basketball game and you can pause your TiVo and
- figure out the probability when the person is making
- that last clutch shot.
- And it could be an interesting experiment for you.
- And actually, I was thinking, an interesting scientific
- experiment, or maybe a high-school science project,
- people have a free-throw percentage and that kind of
- implies that every time someone takes a free throw that those
- are mutually exclusive events, that they're independent of
- the previous time like we said with coins.
- But an interesting idea, you know in basketball people
- always say, he's hot now or he has a streak.
- And so there is this notion and I know I felt it, that there
- are times that you're probability increases or
- decreases, and it tends to be maybe dependent on whether you
- made or missed your previous shot.
- So one thing you might want to-- this is, I think, a
- legitimate science project-- is to either get the data from
- real NBA players and see if they really are mutually
- exclusive events.
- If the probability of making the next shot or the next free
- throw really is independent of whether they made or missed
- the previous one. or whether it actually is dependent.
- Or if you don't have all the data from the NBA or wherever,
- although I suspect you could find it, you could try it with
- yourself and your friends.
- Or maybe since you want to be unbiased, you'll do
- it with your friend.
- You'll see if the probability of them making the next free
- throw really is independent of whether they made the last one.
- Actually, I think that could be quite good and you can get
- quite involved in the analysis.
- So let me finish this up with another scenario.
- We talked about free throws, we talked about flipping a coin,
- and now I'll talk about dice because that is really another
- area where-- well, it's one interesting and you'll probably
- see some problems on probability.
- So, in general, when you're playing games involving dice
- it's always interesting to say, what's the probability if I
- have two six-sided dice, what's my probability of getting--
- I don't know-- a particular number.
- Let's say, what's the probability of getting a 7?
- So to think about that you have to say, well, what has to
- happen for the dice for me to get a 7?
- And I think here it might be interesting to
- draw a bit of a grid.
- So let's say that's my grid.
- Let me split it up into six.
- So let's see, that's splitting it up into one, into two.
- Each of these maybe split it up into three, so
- it would be like that.
- It won't be perfect, but close.
- Like that.
- Like that.
- And like that.
- And let me split it-- you'll see what I'm doing in a second.
- Actually, I didn't want to do that.
- That's good enough.
- Do this, this.
- I'm trying to make a 6-by-6.
- OK, and the reason why I'm doing this is because let's
- make this top axis, this horizontal-- each of the
- situations I can get on the first dice, although I'm going
- to roll them simultaneously, although it doesn't matter if I
- roll them simultaneously or one after the other, or I roll one
- dice one after the other.
- So this first dice I could get 1, a 2, a 3, a 4, a 5, or a 6.
- So this is dice one-- D1.
- And on the second dice, I could get a 1, a 2,
- a 3, a 4, a 5, or a 6.
- And this is dice two.
- And I'm running out of time, so I will see
- you in the next video.
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