Compound Probability of Independent Events Probability of particular sequences of flips
Compound Probability of Independent Events
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- Let's think about the situation when we have a completely fair coin here
- let me draw, I'll assume it's a quarter or something, so this is a quarter,
- let me draw a faint attempt at a profile of George Washington, well that will do.
- It's a fair coin and we're gonna flip it a bunch of times and figure out the different probabilities.
- So let's start with a straight forward one.
- Let's just flip it once, so with one flip of the coin, what's the probability of getting heads?
- Well, there's 2 equally likely possibilities and the one with heads is 1 of those 2
- equally likely possibilities, so there's a one half chance.
- Same thing if we would of asked what is the probability of getting tails?
- There are two equally likely possibilities and one of those gives us tails, so one half.
- And so this is one thing to realize: if you take the probabilities of heads plus the
- probabilities of tails, you get one half plus one half, which is one,
- and this is generally too: the sum of the probabilities of all of the possible events
- should be equal to 1, and that makes sense, 'cause you're taking, you're adding up
- all of these fractions , the numerator and then add up to all of the possible events
- the denominator is always all the possible events, so you'll have all the possible events
- over all the possible events when you add all of these things up.
- Now let's take it up a notch.
- Let's figure out the probability of ... I'm going to take this coin and I'm going to flip it twice.
- ... the probability of getting a heads and then getting another heads.
- The probability of getting a head and then another head.
- There's two ways to think about it .
- One way is just to think about all of the different possibilities:
- I could get a head on the first flip and a head on the second flip,
- head on the first flip, tail on the second flip,
- I could get tails on the first flip , heads on the second flip,
- or I could get tails on both flips. So there's four distinct equally likely possibilities.
- Four distinct equally likely outcomes here. And one way to think about it
- is on the first flip I have 2 possibilities, on the second flip I have another 2 possibilities,
- I could have heads or tails, heads or tails, and so I have 4 possibilities for each of
- these possibilities, for each of these two I have 2 possibilities here.
- So either way I have 4 equally likely possibilities.
- And how many of those meet our constraints?
- Well we have it right over here, this one right over here having 2 heads meet our constraints.
- So this is ... and there's only one of those possibilities, I've only circled one
- of these four scenarios, so there is a one forth chance of that happening.
- Another way you could think about this, and this is because these are INDEPENDENT EVENTS
- and this is a very important idea to understand in probability. We'll also study scenarios
- that are not independent, but these are independent events.
- What happens in the first flip, in no way affects what happens in the second flip,
- and this is actually one thing that many people don't realize.
- There's something called "The Gambler's Fallacy" when someone thinks:
- "If I got a bunch of Heads in a row, then all of a sudden becomes more likely
- on the next flip to get a tails. That is not the case. Every flip is an independent event.
- What happenned in the past in these flips does not affect the probabilities
- going forward. So the probability of getting a head on the first flip
- in no way ... or the fact that you got a heads on the first flip, in no way affects
- that you got a heads on the second flip.
- So if you make that assumption, you could say that the probability of getting
- heads and heads, or heads and then heads is going to be the same thing as getting
- probability ... as the probability of getting heads on the first flip
- times the probability of getting heads on the second flip.
- And we know that the probability of getting heads on the first flip is 1/2,
- and the probability of getting heads on the second flip is 1/2,
- and so we have one half times one half, which is is equal to one forth,
- which is exactly what we got when we tried out all of the different scenarios,
- all of the equally likely possibilities.
- Let's take it up another notch.
- Let's figure out the probability ... and we kind of bit ignoring tails, so...
- let's pace on the tension to tails,
- the probability of getting tails, and then heads and then tails. So this exact series of events.
- I'm saying this exact order: the first flip is a tails,
- second flip is a heads and then third flip is a tail.
- Once again these are all independent events: the fact that I got tails on the first flip
- in no way affects the probability of getting a heads on the second flip
- and that in no way affects the probability of getting a tails on the third flip.
- So, because these are independent events we can say it is the same thing as
- the probability of getting tails on the first flip, times the probability of getting
- heads on the second flip, times the probability of getting tails on the third flip.
- And we know these are all independent events, so this right over here is one half
- times one half, times one half. 1/2 times 1/2 is one fourth,
- 1/4 times 1/2 is equal to one eighth. So this is equal to one eighth.
- And we can verify. Let's try all of the different scenarios again.
- So you could get heads, heads, heads,
- you could get heads, heads, tails,
- you could get heads, tails, heads,
- you could get heads, tails, tails,
- you can get tails, heads, heads,
- this is a little tricky sometimes, you want to make sure you're being
- exhaustive in all the different possibilities here, you could get tails, heads, tails,
- you could get tails, tails, heads,
- or you could get tails, tails, tails.
- And what we see here is that we got exactly eight equally likely possibilities
- We have eight equally likely possibilities, and the tail, head, tail
- is exactly one of them. It is this possibility right over here.
- So it is one of eight equally likely possibilities.
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