Question

a coin is tossed 40 times. find the probability of turning up of heads.

Answer 1

Answer:

Step-by-step explanation:

If the coin is fair and flipped 2n times, the the probability of getting n heads is asymptotic to 1πn√.  That's not the exact answer, but it is way cooler than the exact answer: (2nn)⋅2−2n.

For 40 flips, the approximation gives about 12.61% compared to the exact answer of about 12.54%.  It is impressive that by n=20 the relative error in the approximation is already only about half a percent, and it keeps improving as n increases.

The reason I prefer the approximate solution is that with just a little practice, you can get a good estimate of the answer in your head.  

For example, with forty flips, we need the reciprocal of the square root of 20 pi.  Using 3.14 we need the reciprocal of the square root of 62.8 so the answer is about one eighth, 12.5%.  

For 100 flips, 50 pi is about 157.  The square root of 157 is very close to 121324∗≈12.5.  The reciprocal of this number is 8%.  A Calculator gives an approximation of 7.98% and exact answer of 7.96%, so the ballpark answer that you can get in your head (with some practice) is great!

∗ To get this approximation, we use (1+x)y≈1+xy when xy≪1:

157−−−√=(144+13)1/2=12(1+13144)1/2≈12(1+132⋅144)=12+1324