This is a problem in one of the Khan Academy's videos: The probability of getting exactly 3 heads in 8 flips of a fair coin.
However, I can't understand why the answer is not 3/8, since it is an independent event.
#Probability
wary aurora
lucid haloBOT
This is a problem in one of the Khan Academy's videos: The probability of gettin...
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molten lava
This is a problem in one of the Khan Academy's videos: The probability of gettin...
How did they calculate it?
wary aurora
8C3/2^8 = 7/32
molten lava
8C3/2^8 = 7/32
Okay, to begin with, do you understand the factor of 2^8 in the denominator?
wary aurora
I don't get it. I am confused with this part.
molten lava
How many possible sequences of 8 coinflips are there?
Or equivalently, how many possible 8-digit binary numbers are there?
wary aurora
2^8 = 256
molten lava
Right.
wary aurora
Why is it not 3/8?
molten lava
...why would it be 3/8?
wary aurora
Sorry, I am really confused, because it looks like it's independent event for me
wary aurora
The previous outcome does not affect the next outcome
molten lava
...in what?
wary aurora
err, in any event?
How I understand it is
- There are two possibilities in each flip (Head or Tail)
- Whether this flip is a H or T, there are still two possibilities in the next flip
molten lava
Well, that's not quite what it means.
In probability theory, event A is independent of event B if and only if P(A|B) = P(A).
And you're correct that the coin flips are independent of each other, but I don't see how that leads to your conclusion of 3/8.
wary aurora
Em, flip the coins 8 times, out of the 8, 3 are head?
molten lava
Okay, but like, you're not actually doing any math. You're just saying numbers and then throwing them together in a ratio for no justified reason.
wary aurora
...ok sure
molten lava
Let's think about it like this.
molten lava
What's the probability of getting heads?
wary aurora
1/2
molten lava
And how many do you need?
wary aurora
3
molten lava
And if flips are independent, the probability of getting three heads is therefore what?
molten lava
1/2 x 1/2 x 1/2 = 1/8?
...the left side is correct, the right side is not.
wary aurora
Sorry
molten lava
...is 6 a power of 2?
wary aurora
Sorry
molten lava
And you need exactly three heads, right?
wary aurora
Ya
molten lava
So all the rest need to be tails, which is how many?
wary aurora
(1/2)^5
molten lava
...well, no, it's 5, with probability (1/2)^5.
wary aurora
...ok
molten lava
So the probability of, for instance, flipping three heads in a row and then five tails in a row is (1/2)^3 * (1/2)^5 = (1/2)^8.
wary aurora
Ohhhhhhhhhhhhhh
molten lava
But of course, we need to account for all the various combinations.
We can do this by first multiplying by the number of permutations the eight flips have.
Then we divide by the number of permutations the heads have, and the number of permutations the tails have, since flips with the same result are indistinguishable.
There are, of course, 8! permutations of flips, 3! permutations of heads, and 5! permutations of tails.
And 8!/(3! * 5!) = 8C3.
wary aurora
Wow!
molten lava
This is what's in general referred to as a binomial distribution.
If you have repeated independent trials with a binary outcome, success or failure, and the probability of success is p, and you want exactly k successes out of n trials, the calculation is nCk * p^k * (1 - p)^(n - k).
wary aurora
I understand the concept now. Thank you very much @molten lava 🙇
cedar rockBOT
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wary aurora
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