# SOLVED: A fair coin is tossed 6 times a. What is the probability that exactly two heads will occur . b. What is the mean of this distribution? c. What is the variance of this distribution?

### Video Transcript

so now we ‘re looking at the set of having a coin and and tossing that coin six times. And we want to find the probability distribution and permit ‘s attend at have our perspective beheads. And we know that the probability of Heads is one one-half on a fair coin. So we ‘re tossing at six times, and that ‘s gon na be that by normal rig. And let ‘s let X stand for the issue of successes and that will be getting a heads. And we know that the binomial set has six choose X. We will have our probability of success here, raised to the X office and our probability of a failure, which would be a stern, and that would be due to the six minus X exponent. so in our probability distribution, we ‘re gon na have zero going all the way up to six. And we know that if we have zero successes, that means we will have 6 to 0 and 6 to 0 is one, and then we will have on half to the zeroth ability because we do n’t wan sodium have any successes and we ‘re gon na have one have one half to the one-sixth world power, which means we ‘ll have six tails and we multiply that out. We ‘re going to get one 64th. We wan sodium have one success and we ‘ll have six two ‘s one and 6 to 16 We wan sodium have one tail. Excuse me. One head. therefore, the probability of tales There will be five taels, so one head and five tails and there are six ways for that to be done. So this will give me 6/64 which again reduces down to 3/32 for two. We want a combination of six. Choose two and six. Choose to is 15. You could draw out Pascal ‘s triangle to figure that or, uh, do the calculation with the pictorials or use a calculator and we wan sodium have to Hades. therefore, we will have for tales. And therefore this will become 15/64 and that does n’t reduce down at all. We do n’t have three. And five is the only thing that will divide into their 43 We ‘re gon na end up having our six choose three and six. Choose three is 20 and we ‘re gon na have one half to the third base exponent. so three hats, probability of tales, and we ‘re gon na have three tails. And when you multiply this out, we ‘re going to get 20/64 in 20/60 for both of them are divisible by four. So we ‘ll get 5/16, and then we start to get through. So repeat. so, for four, when we find six, choose four. That ‘s going to be the 15. And we ‘re gon na have the one half to the fourthly baron and one half to the moment exponent. So we ‘ll have four heads and two tails. however, that calculation is gon sodium be the lapp as two. So that ‘s gon na be the 15/64. And for 56 choose five is six and we ‘re gon na have our probability of head. So we ‘re gon na have five of those are probability of tales. This one half we ‘re gon na have one of those, and we ‘re gon na have the 6/64 which is gon sodium be the like just with these, uh, switched around. So we ‘re gon na end up with 6/64 or 3/32 and then having all six of them being Hades, we ‘re gon na have our six choose six, which is one our probability of heads is one one-half we ‘re gon na have six of those. Our probability of tales. This one one-half. But we ‘re not gon na have any of those. And this is gon sodium be the same. calculation is this, and we ‘re gon na have won over 64. So there ‘s our entire probability distribution for tossing a coin, a bonny coin six times.

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