00:01
We want to find the probability of getting between four and eight heads in 12 tosses of a fair coin.
00:08
Okay, so we have, for part a, the binomial.
00:11
The probability of getting heads is 0 .5, and we have 12 trials.
00:17
So for this bit, we're going to use the binomial formula, which is that the probability of exactly x successes is equal to n choose x, p to be x, 1 minus p to the n minus x.
00:32
And between 4 and 8, if it's inclusive, it means we need 4, 5, 6, 7 and 8.
00:45
Any of these, so we have to add them up.
00:48
So this is going to take a little bit of time.
00:50
I'm going to write out what we'll put into the calculator and maybe not flip it upside down.
00:59
There we go.
01:00
So before, we have this term here, which is given as n factorial over x factorial and minus x factorial.
01:12
So it's n choose x, here it's 12 choose 4, and that's just the combinations.
01:17
How many ways you have of putting all of these events in order? so 12 choose 4 is 495 times 0 .5 to power of 4 times 0 .5 to power of 8, so it's just to the 12.
01:37
All of these are going to be 0 .5 to power of 12, just because p and 1 minus p are the same thing here.
01:49
Then we have 12 choose 5, which is 792, 12 choose 6.
02:00
So this is a beautifully symmetric binomial because it's following the symmetry of this function, like so.
02:11
Okay, let's add these up, and then i'll know how many 0 .5 to the power of 12s we have.
02:23
So the answer to this bit is 3 ,498 times 0 .5 power of 12, or as a decimal, that is 0 .84540.
02:45
So that's the first part using the binomial distribution.
02:51
Next we're going to use a normal approximation.
02:54
So a normal distribution has two parameters that we need.
02:57
So this is pi.
03:00
We need a mean, and we need a standard deviation.
03:04
And we get these from the binomial.
03:06
So the mean of a binomial, the expected value, is equal to np.
03:10
So that's going to be six.
03:12
The standard deviation is the square root of the variance, which is n p, 1 minus p.
03:19
So we have root 3, which i'm just going to keep in that form for now.
03:26
So now we have our parameters.
03:35
We just need to use the normal distribution.
03:40
So we want between four and eight.
03:52
We need to do something called continuity correction here.
03:56
And that's because binomial is discrete...