3-4-4. Probability

And, yes, the answer is zero. And we find a little law here we just want to point out, which is the probability of heads plus the probability of tails equals 1. And the reason why that’s the case is the coin either comes up heads or tails. There is no other a choice. So … Read more

And, given that, what number would you now assess the probability of tails to be?

And the number is 1. That’s the same as 100%. 1 just means it always comes up in heads.

A loaded coin is one that comes up with one of the two much more frequently than the other. So, for example, suppose I have a coin that always comes up heads. What probability would I assess for this coin to come up heads? What would be the right number over here?

And I would say the answer is 0.5. Let me now go to a coin that is what is called “loaded.”

Let’s talk about a fair coin. The probability of the coin coming up heads is written in this P notation. This reads probability of the coin coming up heads. And in a fair coin, the chances are 50%. That is, in half the coin flips, the coin should come up heads. In probability we often … Read more

You now have a basic understanding of probability. Great job! Let’s quickly summarize what was covered in this lesson. You learned about the probability of an event. Such as the outcome of a coin flip. You learned that the probability of the opposite event is one minus the probability of this event. And you learned … Read more

And once again, we can answer this using a truth table. Now the truth table will have 36 different entries, six for the first throw times six for the second throw, and there isn’t enough space on this tablet to draw all the 36 entries. So, let me just draw the ones that really matter, … Read more

Suppose we throw a fair die twice. What do you think the probability of a double is? Double means both outcomes are identical with the same number regardless of what that number is. The actually an important number because in many games involving two dice, have different rules when these come up with the same … Read more

In truth table-speak, there are 6 outcomes, 1 to 6. Each has the same probability over six. Half of those numbers are even–2, 4, and 6, so if we add those up, we get 3 1/6–the same as a half. The outcomes is 0.5. Now I’m finally going to make, as my final quiz, a … Read more

So let’s do one final exercise. Now I am throwing dice. The difference between dice and coins is that there are now 6 possible outcomes. Let me just draw them, and say it’s a fair die, which means each of the different sides comes up with a probability over 6 for any of the numbers … Read more

And my answer is 0.288. How do I get that? Let’s look at the three critical cases. H T T is 0.6 for H times 0.4 for tails times another 0.4 for tails again and it gives me 0.096. Now it turns out this case over here has the same probability because all we do … Read more

And you can debate it, but I think the best answer is no. This is what’s called a fair coin, and that means it really has a 50% chance of coming up tails. So let me spin it again. [sound of coin spinning] And, not surprisingly, it actually came up tails this time. So probability … Read more

Now that was a challenging question. I’m going to make it even more challenging for you now. I’ll give you a loaded coin–the probability for H is 0.6. I expect this will take you awhile on a piece of paper to really calculate this probability over here. But you can do exactly the same thing. … Read more

And this answer is tricky. We will derive it through the truth table. Now there’s eight possible cases. Flip one can come of heads or tail; same for flip two, heads, tail, heads, tail; and the same for flip three and if you look at this every possible combination is represented. For example, these are … Read more

Let me now make it really, really challenging for you. I take a fair coin and flip it 3 times, and I want to know the probability that exactly 1 of those 3 flips comes up heads.

And, yes, it’s in the second case and in the third case. The extreme cases of heads, heads and tails, tails don’t satisfy this condition. So the trick now has been to take the 0.25 probability of these two cases and add them up, which gives us 0.25 + 0.25 = 0.5. This is the … Read more

Given that, we now have to associate a truth table with the question we’re asking. So where exactly is, in the outcome, heads represented once? Please check the corresponding cases.

And the answer shall be 0.5. And this is a nontrivial question. Let’s do the truth table. So, for flip-1, we have the outcomes of heads, heads, tails, tails. For flip-2, heads and tails and heads and tails. These are all possible outcomes. And we know for the fair coin each outcome was equally likely. … Read more

The truth table gets more interesting when we ask different questions. Suppose we flip our coin twice. What we care about is that exactly one of the two things is heads, and thereby exactly the other one is tails. For a fair coin, what do you think the probability would be that if I flip … Read more