Frequency Probability and Unfair Coins A different way of thinking about probability when we don't have equally likely events
Frequency Probability and Unfair Coins
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- So far we've been dealing with one way of Probability, that was the probability of (A) occur,
- The number of events that satisfy A over all number of the equally likely events
- All of the equally likely events.
- And so in the case of a coin, The Probability of heads was a fair coin
- With 2 equally likely events, satisfies being "Heads"
- So there's a half chance of having a "Heads"
- Similarly for tails, if you took a die
- The probability of getting an even number.
- Well there's 6 equally likely events, and there are three even numbers you can get
- Which are, two, four and six
- So there's three over six, or simply,
- One half chance
- And this is a really good model
- of equally likely events. Now I'm going to change things up a little
- I'll draw a line, this was just one way about thinking of probability
- Now we're going to talk of probability where we can think of equally likely events
- And in particular I'm going to set up an unfair coin
- So this right over here, is supposed to be my unfair coin.
- One side of that coin is heavier than the other, So this is the head's side
- And obviously on the back, is Tails
- As I said, this is an UNFAIR coin
- Now I'm going to make an interesting statement about this unfair coin
- It doesn't fit into the model I've created above,
- And this interesting statement is, above we had a 50-50 chance of getting heads
- I'm going to say, here, that the probability of getting Heads, for this coin right here,
- is 60% (Sixty Per Cent)
- Or another way to say,
- 0.6 or 6 upon 10.
- Or another way to say this is 3/5 or three-fifths
- Now we can't say there are two equally likely events
- There are two possible events
- The coin may fall on the heads, or tails
- So it is obvious that either we'll get Heads or Tails
- But there aren't really two equally likely events
- Counting the number of events that satisfy, over all the possible events
- In this situation, we can visualize the probability.
- This is fundamentally different from the model we talked about above.
- We have to kind of take, a frequentive approach
- Think about it in terms of frequency,
- And the way to conceptualize, 60% chance of getting heads
- If we had a super large number of trials. This means if we are to flip this coin
- Like a zillion times
- We would expect, 60% of those would come up as heads
- So I'm clear how I determine, that this is60%! Maybe I ran a computer stimulation, maybe I should know exactly all about physics
- Of this, and I can completely marvel how it's going to fall everytime
- Or maybe it's like, I have a tonne of trials, I flip the coin a million times And I said, well, 60%
- Of those, therefore, came up heads.
- And we can make a similar statement about Tails!
- So, the probability of Heads is 60%
- The Probability of Tails, well there're only two possibilities, whether there can be heads or tails!
- So if I say the probability of heads or tails, is going to be equal to 1 !
- Because there also can be a chance of getting hundred percent Heads or tails
- And these are mutually exclusive events, you can't have both of them
- So this is going to be, the probability of tails is, going to be 100% - Probability of Heads
- which of course is 60% (as above)
- Which is equal to 40%
- or as a decimal, it is 0.4
- Or as a fraction, four tenths, or Four upon Ten, or in its simplest form, it is Two Fifths, 2 over 5
- So once again, we can't call them equally likely events. If we toss a coin a zillion times, and we get
- 60% of the coins as heads, it is obvious that the rest coins have to be tails
- It's time for some problems now,
- The probability of getting heads on our first flip and heads on our second flip.
- So once again these are independent flips. The coin has no memory, regardless what I got on the first
- flip, I have equal chance of getting heads on the second flip. I can get heads and tails on the first
- as well, so the probability of heads on the first flip X Probability of getting heads on the second flip.
- Now we already know that the probability of getting heads on the first flip is 60%
- Or let's write it in the decimal form, it is 0.6
- Also, P(H2) is 0.6. And we can just multiply. I'll do it right here, so this is 0.6 x 0.6
- It is equal to 0.36.
- We're taking 60% of 0.6. 0.36, or another way to say it that we have 36% of getting two heads in a row
- Given this unfair coin, remember if it was a fair coin, one half times which is one over four which
- 1/4 or 25%.
- Now let's see a slightly complicated example;
- Let's say, the Probability of getting a tails on the first flip, getting a heads on the second flip,
- And getting a tails on the third flip. So this is going to be equal to the probability of getting a tails on the first and third flip, and a heads on the second
- It is equal to The Probability of Getting Tails on the First flip, multiplied by the probability of getting
- heads on the second flip, heads being on the second time doesn't affect the probability of heads and similarly, a tails, again, on the third flip
- And we know that the probability of getting a tails is 0.4
- The Probability of getting a heads(on any flip) is 0.6 and the probability of getting tails on the third flip is 0.4
- So once again we just have to multiply all these probabilities,
- So 0.4 X 0.4 X 0.6 = 0.096
- Or another way it to write is 9.6%, a little less than 10%. We have a 9.6% chance of getting this probability
- Remember, in this case, the position of heads or tails, first, second or third does not affect
- Their probability
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