Coin Flip calculation...

Question

It is desired to perform an experiment with an honest coin (a coin in which the chances of being heads are the same as of being tails). To do this, the coin is tossed 64 times. Assuming that these 64 tosses are a sample of the multitude (infinite) of tosses that we can make, calculate a probability that in 64 tosses it will occur:

A) At least 40 heads

My answer for this is 0.02997, or 3%. The problem is I can't apply the calculation for Excel/Sheets using a formula, which is needed.

B) between 24 and 36 heads.

C) Less than 28 heads

Answer 1

standard deviation = sqr(npq) = sqr(64)(1/2)1/2)) = 8/2 = 4

standard deviation = 4. n=64, p=1/2, q=(1-p) = 1/2

mean = np = 64(1/2) = 32

40-32 = 8

8/4 = 2 standard deviations from the mean

for more significant digits use an on line calculator or z tables

easiest solution is use the 'empirical" rule, 95% fall within 2 standard deviations.

half of 5% fall above 2 standard deviations = 2.5%.

Or use z tables for z=2. 2.275% fall above 40

24 to 36 is 2 standard deviations below and 1 above the mean

with the empirical rule 68% fall within 1 standard deviation from the mean

that's 34% from 32 to 36, and 95/2 = 47.5% between 24 and 32.

add them: 34+47.5 = 81.5%

28 to 32 is one standard deviation below the mean: 34% less than 28 is 16% or 15.87% with z tables or an on line calculator