SOLVED: What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up tails?

Video Transcript

Okay, so he was trying to figure out we flip a fair mint, the number of flips that would be attained, he said. A act of flips that would be obtained either until two tails are achieved or until the coin has been flipped six times. It ‘s well ignite xB the number of queen flips. And then today, clean coin. The probability of getting a heads is adequate to the probability of getting eight Hales, which is 1/2. so do n’t do this in stages, right, because we do n’t have excessively many cases to look at. The number of folks is a minimal of two, and it goes up to six. So we ‘ll look at what happens when ten equals two when ten equals 345 and six and then to get out from there. So the first case, it ‘s gon sodium be the probability of when X is equal to two eso This assumes and that our 1st 2 flips for both tails and that occurs with the probability of it entails times the probability of getting tails. Or we could good square it. This would be 1/2 times 1/2 or point 25 Okay, so then The second lawsuit is going to be where we have three coin flips. And indeed the thing is, in this example, we know the third gear football details. But one of the 1st 2 is the heads. The club does not matter. And since rate does n’t matter, we will use a combination. So a combination, uh, to choose one because we ‘re choosing one of the 1st 2 to be heads multiplying that, you know that there are in these three flips there ‘s two tails. So the probability of tales times credibly tails and there is one mind, so we want to multiply it by the probability of getting heads. You ‘ll see that the experience here there is a two and a one is equal to the number of flips, right ? Yes. Once we types that into our calculator but ends up being to choose one. This these problems are the same. It ‘s actually getting 10.5 to the third. Karen ‘s gon na be equal to 0.25 master a newly page here. so for the probability that we have four flips, he ‘s gon na follow this same recipe. We ‘re gon na have to choose a combination again or is n’t count. But two of the 1st 3 foots were heads or choosing to from three to be our heads. Two of these four is going details. We ‘re stopping at two tails. Okay, so you want the probability of Tales Square ? But the early two are going to be heads actually wanted ticket, probably a head squid. And again you see the exponents two and two is equal to four that this will be equal to three Choose to That was 1/2 to the fourth and this ends up being equal to we put it in the calculator 0.187 five. So there ‘s two more cases. Nest is where we have five flips and following the like operation in yet choosing three from Forbes. We ‘re choosing three of the first four flips to be heads, knowing that the 5th 1 was tails And then since we have two tails will multiply that by the problem of getting entails squared. But then we besides have three heads. sol I want to point out that it probably getting our heads three times. Where to the third power ? Yes, this is it being for choose three times 1/2 to the fifth, which is adequate to point 12 five. Yes, and we have one case left to look at. And that ‘s going to be where we have six flips, right ? therefore careless of what happens on the sixth throw, we start flipping. Okay, but four or five of the 1st 5 flips has to contain heads. Eso basically there could be at most one. And thus besides embrace that we have to add these two cases together. So we have the lawsuit where we ‘re choosing five. Choose four with one has one tail and the foreheads. Or there ‘s the event we ‘re choosing 55 Where ‘s all heads and no Kale ‘s. Okay, so there ‘s either case, whether it was one tail just this first separate or the case where there was no tales in the second function. And so we add that up together. sol get five shoes for okay times 1/2 to the fifth and leaving it at that to five. Choose five, 1/2 to the one-fifth. And, of class, whether the last one is tails or not, we are stopping. So it it is a relevant approve, and this is equal to 0.18 seven ticket and cement. We figured out the artillery for each lawsuit, we can move on to find the have a bun in the oven value. Okay, so the expect value then is going to be equal to the count off each. Tom had tails happening times the probability of that currents. then when we have the lawsuit where ten is equal to two for two flips, that probability was, uh, 0.25 I ca n’t allow that to. When we had three flips slender, the probability was besides 0.25 for four flips got more matter to. It was 0.1. Eat 75 for five. Flips are probability was 0.1 to five. And then, last, to get six flips, credibly of that occurring was 60.18 75 Okay, thus once we add all that up, we ‘ll find that the expected value is about three point 75 So you ‘d expect to get between three and four flips to get your two tails, but closer to four

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