What you have there is a way to calculate probabilities.
t= probability of event occurring (p)
1-t = probability of event not occurring (q)
The Brackets with n over i is pronounced "n choose i". You will find a button on your calculator (if its scientific) looking like nCr.
nCr = n!/(r!*(n-r)!)
n! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9....... x n
B(t) means Function B in terms of t. But B really stands for Binomial, as this function is a Binomial expansion.
(a+b)^n = E(nCr)*(a^(n-r))*b^r
That big E you see in your picture is Called Sigma, it means "Sum of terms". The P's don't play a part in the formula but just are there to tell you which term it shows. so P0 is the First term.
So if P0 then : (n choose 0) * (a^(n-0))*b^0
which equals: a^n.
The way you get the numbers like 5 10 etc, is as said before. It is pascals triangle. Which creates the sequence of numbers, instead of calculating (nCr)
so if I had (a+b)^5, I could say (5C0)*stuff + (5C1)*stuff + (5C2)*stuff + (5C3)*stuff + (5C4)*stuff + (5C5)*stuff
If you calculate it, 5C0=1 5C1=5 5C2=10 5C3=10 5C4=5 5C5=1
But you can bypass calculating these numbers individually by creating a pascals triangle. You do this by starting with 1 (when n=0) and start a new row (n=1) add the term above the term you're calculating to the number before the number above the number you're calculating (Hang on what?) I'll try to demonstrate
n|
0| 1 <n=0 only one term of 1
1| 1 1 <n=1 first number (0+1=1) second number (1+0=1)
2| 1 2 1 <n=2 second number (1+1=2) etc...
3| 1 3 3 1
4| 1 4 6 4 1
5| 1 5 10 10 5 1 < As you can see this is the same sequence when you calculate 5C0, 5C1, 5C2, 5C3, 5C4, 5C5
Hope this helps and i haven't over complicated things.
Oh and tE[0,1] means t can only lie between 0 and 1 (But that is only for the probability binomial expansion)
In:
(a+b)^n = E(nCr)*(a^(n-r))*b^r
a and b can be any number.
Damn! wasn't quick enough.
Its finally here!
