SOLVED:(2) A coin has 1/4 chance of having heads; 3/4 chance of having tails It is tossed n times a) What is the expected number of coin tosses needed to get heads? b) What is the probability that the

Video Transcript

Okay, so we ‘re flipping 5 coins and we ‘re going to be finding some different probability based off of that. So these are going to tar a little act easier in terms of victimize structural ze, then we ‘ll kind of work our direction to the harder ones sol, first gear 1. I think about all heads and the probability of flipping a head. A convention coin is 5051 half. We have this probability of 1 half that ‘s on the beginning coin. Each coin is going to have each time we flip it. I should take going to have a probability of 1 half or 4 heads to get a head 5 times in a row. We would just multiply all these together : half 1 one-half one-half 1 half 1 half and we multiply these. We get 1 over 32 point. That ‘S the probability of doing so now consider some unlike cases here we want to get at lyons, let ‘s think it possibilities. If we want 1, we could look ahead in the first gear 1 and have all the or we could get. The second base 1 have a fit lines or we could flip a third base 1 or the one-fourth 1 or the fifth 1. So there are 5 possible instances of flipping 1 head and the rest, and so we ‘re going to find and in our designate total number of possibilities to possibilities for each coin slips multiplied in concert. These 32 possible combinations as it the probability with 1 pass, is 5 on the third. We talk about the 5 different ways as in the foremost 1, as on the second, as on the third has on the fourth, as on the fifth and with those combinations, the rio de janeiro, they only find out a 3. We quote pitto our tables. now this is so we ‘re gon sodium of its value common, for it is finding how many ways to those 2 spaces out of 5, iodine ‘m going to be in any kind of a oreolet ‘s draw 5 again to tawell. Flip 1 and 2 could be tails, we ‘re fair assuming the form for we have 1 and 31415, then we can have 2 and 32 and 42 and 5 its coil down and get all these possibilities, and here we have quite a count of them. I ‘M sorry, you got back here apologizing. Why merely come back. I actually went to imitate way. You see what i ‘m doing a far as picking the spots. It is kind of a topic of listing all compito. here you go 12131415. We have 2324253435 and then 45. So these are the spots where they are you ‘ll comment. There are 10 different places, 10 different instances. We get have precisely 2 hats in these positions and we talk about the 3 possibilities. We have a tent and we can simplify that liteiny nowadays at least 1 tail. Okay, so at least seems we not 1 taille all the way up here, sometimes when we want to do complimentary trouble, where we do n’t account how many times we can get something you want to count, how ca n’t get something here, it ‘s going to be easier to do That, because at least 1 means the alone thing we do n’t want to go, take it, so that would mean we have 5 mind and there ‘s alone 1 way of doing it. Look at 5 times we get ahead each time. indeed this is our part a hera. That ‘S the only alone possibility we want so what we would do for complimentary 1. What we do n’t want- 1 minus 1 over 3 to 1 is 3 over 32 point, so we subtracted and we get 31 over 3. So we have identical good odds for you want to have recha. If we flip a 5 point, we get a only prospect entirely room we ca n’t be fiall right. so, let ‘s go down a little snatch here, we ‘ll get to the last accommodation, at least to hedwe ‘re, going to apply something to do complimentariehere or we to think about for one ‘m going to use complimentary. So what i do n’t want is 0 heads, there ‘s only 1 way of doing that or if i get 1 steer. So if i have 1 heading that could keep the first gear position. The second situation, a third base position, the one-fourth on this idea, so the rest of these would be table holloway through a torto get the theme. So all of these would be tales. There are 6 possibilities where this does n’t work. We have 1, where we have no head. We have 5 would not have 1 head, so our probability would be 1. Minus v. Thirwhich is 2630. I raise the space here indeed we ‘ll find this, which is 13 out of 16, therefore feel pretty well on sutton 16 prospect. That happened all veracious now. This last 1 on this 1 will be precisely 3 tails. We know the last flip to tawell that ‘s 18. Before this we have what we know. This is a tail if we have 3 and full precisely 3. We know that 2 of these have to be as so. We can do something like what we did earlier with taking about vomit the first and second details for third base, the first and fourth second and third base, second and fourth, third base and 4 : p 123456 comb. now this is where this is catchy kind of think about this a little sting, it ‘s not going to be out of 32 poistwell. In this case we have 6 possibilities, but it ‘s truly entirely for these 4. This one ‘s fixisti shakes there ‘s nothing. We can do about that. so actually there ‘s merely 16 possibilities in this case, so it ‘s kind of poundtothis, 1 being what it is means. We alone are in truth focused on the beginning 4, so there ‘s 16 ways. now we can rule those and there ‘s 6 possibilities that were for the 6. Indosixteen is 3, so we ‘ll kind of do a flying, recancel everything hera and but the idea is think about how many possibilities are on to how many of them are what you would think, as winning combination you ‘re playing a bet on in this. Your winner, how many ways to info the beginning match are barely the first 1 at least is more fit ahead. then you kind have to think about the number of possibilities that work for you sometimes you ‘re, going to have to use this theme of complimentary probability where we have 1, where we do n’t win subtract that from 1 and sometimes you ‘re going to have to set up some Things where the sample faith, the total of possibilities shrink, because we have to information about some of the rules already. So hopefully, this kind of avail in some things for wantin.

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