Binomial Distribution

Let’s now make you think really hard. In 5 coin flips, how many outcomes will you have 2 heads. This is a serious and non-trivial question.

The answer is 5. There’s 5 different ways in these 5 outcomes. To place heads–could be first, second, third, fourth or fifth. So these are 5 different ways.

With 5 coin flips, how many outcomes will have exactly 1 heads, hence 4 tails.

And the answer is 0. With an odd number of coin flips, one has to be more than the other. There’s no other way. Okay. I tricked you a little bit.

Now. Let’s go to 5 coin flips. How many outcomes have the same number of heads and tails? This is a trick question.

In going through the truth table, there’s a bit more info. So let’s find the one where the number of heads and tails are the same. Those three on the left side and those three on the right side for a total of 6.

In this lesson, we found the probability that a coin would land on heads k times out of n flips. If the probability of heads for the coin is p, we get the following formula for the probability that the number of heads will be k. The first half is n factorial over n minus … Read more

And the approximate answer is 0.236. That’s the probability of exactly 9 heads out of 12 coin flips for this heavily loaded coin that mostly gives you heads. And again, the answer is 12!/12-9. This is 3! 9!. Then we have to compute the probabilities being (0.8)⁹ and 1-0.8 is 0.2³ and that is the … Read more

Let’s now go to 4 coins and ask the same question.

Now, you’re ready for the real challenge. You flip the coin 12 times in the care about how likely it is to get heads 9 times out of the 12. This is not a trivia question, but you should be able to get it right.

And here is my answer 5!/3!*2! is 10, and we have 3 heads, so we put 3 in here and 2 tails, so we put 2 in here. Putting these all together gives us 0.2048, which is half the probability of the previous question.

So, this does not go and get the same for 3 heads. I leave this here, but obviously, the numbers aren’t correct anymore.

So interestingly enough, we can do the trick as before 5!/4!*1!, which is the number of outcomes at exactly 4 heads and we know that’s 5. Four heads means one tails. There’s five ways to place one tails and this one over here. The question now what’s the probability of those? Well, they have heads … Read more

Let’s now say for the same loaded coin, we flip that coin five times with that same load of probability over here and we care about the out of five times heads comes exactly four times. Now, I want you to compute this probability over here, It’s a completely nontrivial question. If it’s too hard, … Read more

So here is my truth table of these eight different outcomes. The ones that has head exactly once are this one, this one, and this one, but they’re not all equally likely. Heads, heads, heads is much more likely than say tails, tails, tails because heads has a probability of 0.8^3. This one here has … Read more

Now, I’m going to make it really difficult. I’m going to give you a coin–let’s call it loaded. So, the probability for heads will now be 0.8 and therefore the probability for tails is 0.2. To make it easier, assume only a 3 coin flips and ask the probability of heads coming up exactly once. … Read more

And the answer is 5!/5-3 is 2!/3! and that gives us 10. 10/32 gives us the probability of 0.3125.

Let’s now modify it and ask, what are the chances it becomes a head three times? What’s the probability for that to happen?

Now, we know from our previous consideration that there are five ways in which the number of heads could be one. 5!/4! 1! happens to be 5. We also know that there are 32 possible outcomes. It is 2⁵=32 outcomes. This is the size of a truth table. So, 5 out of 32 outcomes has … Read more

And the answer is 2. If you look at the truth table head-head, head-tail, tail-head, and tail-tail—these are the four possible outcomes. Those two outcomes over here yield an equal number of heads and tails.