Coinflip Probability Question

Question

What is the Probability of getting 60 more heads than tails in no particular order, at any time, in 1000 total flips.

And to go with that what would be the probability of same question with 10,000 total flips, and 100,000 flips.

I can figure out the probability of getting at least 530 heads out of 1000, but I can not figure out the Probability of it happening at any time (in other words the 60 heads more than tails could happen right away, in the middle, at the end, or at any other point) I want to know the chance of having 60 more heads at any point with 1,000 flips, 10,000 flips, and 100,000 flips.

I know this is asking a lot, but if you have the time I would also be very interested in the same question but the probability of getting 30 more heads than tails in no particular order, at any time, in 1000 flips, 10,000 flips, and 100,000 flips

Also any explanation would be appreciated. Thank You Very Very much for any answers.

Answer 1

h = 60+ t

h+t = 1000

60+2t = 1000

2t = 940

t = 470

h = 530

n=1000

p = 1/2

q = 1/2

npq = variance = 1000/4 = 250

sqrnpq = standard deviation sqr250 = 5sqr10

np = mean = 500

(530-500)/5sqr10 = z = 30/5sqr10 = 6/sqr10 = 6sqr10/10 = 3sqr10/5 = about .6(3.162= about 1.9

= about 2, by the empirical rule, 95% fall (within 2 standard deviations from the mean with

(1-.95)/2 = .05 = .025 = 2.5% above z= 2

slightly less than 2.5% probability of more than 60% heads in 1,000 flips of a fair coin

using a normal distribution as an approximation to the binomial distribution