Binomial Distribution

And let’s really ask about probabilities. Let’s say for now, we have a fair coin with a probability of heads is 0.5. If I flip a coin five times, what’s the probability the number of heads is exactly 1. You should be able to compute this.

So, I get 317,750, and the logic is I just plug in these numbers, 125!/122! 3!. When I evaluate this, I find that these guys can be easily reduced to 125*124*123. From 122 on, these factors exactly cancel. 3 expands to 3*2*1 also known as 6. When you divide these two things, you get 317,750.

The key observation is that this thing over here is n!/(n-k)! and to see this let’s go back to the case where we had 4 heads. In the 4 heads case, we multiplied all the way to times 5 over here. We only multiplied all the way to 7. These are the 4 heads that … Read more

And this one is the correct one and I have to admit I made a mistake a little bit by taking a symmetric example where k is exactly half of n.

So let me give you the mathematical formula that you might be familiar with. n! for any number n is the same as n(n-1)(n-2) all the way to times 1. Let’s call it factorial. So 10 factorial for example will be 109876 and so on. If you look at this equation over here, I’ll give … Read more

I was given 10*9*8*7*6 which is 30,240 with a pick of those five heads and we’re over counting by 5*4*3*2*1 which is 120 and this gives us as a quotient 252.

Let’s say I have 10 coins. You just get 10 of these shiny silver coins here. So we got 10 coins. And I want about 4 heads in these 10 coins, and just apply the same logic to find an answer. The first time I’ll do it for you. I can place those heads in … Read more

And that log is 10 again. There’s two proofs. One is I can just flip heads and tails. So three heads means two tails. I can do the exact same game as before where I placed tails as opposed to heads and it gives me the same equation as before, but let’s do it the … Read more

Let’s now go to 3 heads if you think the result is no.

And the answer is 10 and this is a non-trivial answer. So you could go and place the first heads anywhere in these five elements–say here– and there’s five different ways to place heads. You can now place the second heads among the remaining four–for example, you could place it over here and it gives … Read more

So we talked about coin flips, and we flipped some coins. Now, I want to flip many coins including this 2-dollar coin whose country of origin I just don’t remember. Perhaps you can post on the forum where this 2-dollar coin might be from. So let’s ask an easy quiz. Suppose we do 2 coin … Read more