Multiplication rule for dependent events
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Dependent probability example
Let's do another one of these dependent probability problems. You have 4 coins in a bag. 3 of them are unfair in that they have a 45% chance of coming up tails when flipped. The rest are fair. So for the rest of them, you have a 50% chance of tails or a 50% chance of heads. You randomly choose one coin from the bag and flip it 4 times. What is the percent probability of getting 4 heads? So let's think about it. When we put our hand in the bag and we take one of the coins out, there's some probability that we get an unfair coin. And 3 of the 4 coins are unfair. So there's a 3/4 probability that we get an unfair coin. And then there is only 1 out of the 4 coins that's fair. So there was a 1/4 probability that I get a fair coin. So unfair, let's remind ourselves-- an unfair coin has a 45% chance of coming up tails. So this means that I have a 45% chance of tails, which also means-- and we have to be careful here because they're asking us about heads-- if I have a 45% chance of getting tails, that means I have a 55% chance of getting heads. I have a 100% chance of getting one of these two. If it's 45% for tails, 100 minus 45 is 55 for heads. For the fair coin, I have a 50% chance of tails and a 50% chance of heads. 50% heads. Fair enough. Now I want to know, in either of these scenarios, what is the percent probability of getting four heads? So given I've got the unfair coin, the probability of getting four heads is going to be 55% for each of those flips. So the probability of getting exactly four heads is going to be 0.55 times 0.55 times 0.55 times 0.55. And so the probability of picking an unfair coin and getting four heads in a row is going to be equal to 3/4 times all of this business over here. So that's 3/4 times-- and this is 0.55 times itself four times, so I could write it as 0.55 to the fourth power. And we'll get the calculator out in a second to actually calculate what this is. Now let's do the same thing for the fair coin. If I did pick a fair coin, the probability of getting heads four times in a row is going to be 0.5 times 0.5 times 0.5 times 0.5. Or the probability of getting the fair coin, which is 1/4 chance, times the probability-- and getting four heads in a row is going to be 1/4 times all of this. So it's going to be 1/4 times-- this is just 0.5 times itself four times, so that's 0.5 to the fourth power. So let's get the calculator out to calculate either one of these. So we get 3 divided by 4 times-- and it knows that when I do the multiplication, it's not in the denominator here. So it's 3/4 times-- and I'll just do it in parentheses, which I don't have to do it in parentheses, because it knows order of operations-- so 0.55 to the fourth power is equal to 0.-- let me write it down. Let me take it off the screen so I can write it down properly. Actually, let me just do both of these calculations. So this probability is that one right over there. And then this one down here is 1 divided by 4 times 0.5 to the fourth power. So it's equal to that right over there. And so let's be clear. The probability of picking the unfair coin and then getting four heads in a row is this top number. It's like roughly 6.9 chance that you get the unfair coin and then get four heads in a row. The probability that you get the fair coin and then get four heads in the row is even lower. It's only a 1.6% chance. Now the probability of getting four heads in a row either way is going to be the sum of this and this, or the sum of that and that, which is going to be-- let me keep my calculator out-- so it's going to be equal to-- I can just take the previous answer. Let me just retype it so I don't confuse you. So 0.015625 plus 0.0686296875. I'm going to round it anyway, so it won't matter too much. So if I take the sum-- let me take this off screen so I can still see it and then let me write it. So what I got here, this one is 0.068629. And I'll round it, 7. This down here was 0.015625 and when you add these two up-- because we just care about getting four heads either way. There's a probability of getting it this way with the unfair coin. This is the probability of getting it with a fair coin. We want it either way. So let's add the two, which we already did on our calculator. So if you add that number to that number. You get 0.08425 and it keeps going, but I'm just going to round it. This is equal to 8.425%, or if I want to round it a little bit more, 8.43% chance of getting four heads in a row. And once again, that's a slightly higher number than if all of the coins were fair. Because there's a 3/4 chance that I get a coin that has a better than even chance of getting heads. So that's why this number is going to be a little bit higher than the probability if I had a fair coin, of just getting four heads in a row.