Transcription
Let's do another one of these dependent probability problems. You have four coins in a bag; three of them are unfair, and they have a 45% chance of coming up tails when flipped. The rest are fair, so they have a 50% chance of tails or a 50% chance of heads. You randomly choose one coin from the bag and flip it four times.
Let's figure out the probability that we get an unfair coin. Three of the four coins are unfair, so there is a 3/4 probability that we get an unfair coin. There is only one fair coin out of the four.
Now, given that I have an unfair coin, let's remind ourselves: an unfair coin has a 45% chance of coming up tails. This means that I have a 45% chance of tails, which also means 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. Whatever I do, I have a 100% chance of getting one of these two; if it's 45% for tails, 100 - 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 if, 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 * 0.55 * 0.55 * 0.55. 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 0.55 to the fourth power. We'll get the calculator out in a second to calculate what this is.
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 * 0.5 * 0.5 * 0.5, or the probability of getting the fair coin, which is 1/4, times 0.5 to the fourth power. Let's calculate either one.
So we get (3/4) * 0.55 to the fourth power. 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 this down here is 1/4 * 0.5 to the fourth power, so it's equal to that right over there.
To 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 a 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 a row is even lower. You only have this and this sum of that and that, which is going to be—I'll keep my calculator out—going to be equal to—I can just take the previous and let me retype it so I don't confuse you—0.0625 + 0.00390625.
Let me retype it. I can see it; let me write it. So what I got here, this one is 0.068629, and round it down to 0.0625. When you add these two up, we just care about getting four heads either way. The probability of getting it this way with the unfair coin is this; this is the probability of getting it with the fair coin; we want it either way, so let's add the two, which we already did on the 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. So this is the same thing as 8.425%, or, if I want to round it a little bit more, an 8.43% chance of getting four heads in a row. And once again, that should be—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. That's why this number is going to be a little bit higher than the probability, with a fair coin, of just getting four heads in a row.