If you flip a coin 44 times what is the best prediction possible for the number of times it'll land on heads

12% chance of 22

35% chance of 21, 22, or 23

55% chance of 20, 21, 22, 23, or 24

How can 22 be a good prediction when it's wrong 88% of the time?

I'd say from 20 to 24 is a good prediction without being trivial.

Step-by-step explanation:

C (n, k) = n Choose k = n! / (k! (n-k) !)

aka binomial coefficient

aka from n things choose k

To get probability of getting heads 22 times in 44 tries, you divide number of ways to get heads 22 times by number of ways to assign heads or tails to each throw.

Two ways to assign H or T to first, times two ways for second throw,

gives 2^44. That's the denominator.

Number of ways to get heads 22 times is the same as number of ways to choose which 22 flips of 44 are to be heads, or C (44,22)

To get 12% calculate C (44,22) / 2^44

To get 55% calculate C (44,20) / 2^44 + ... + C (44,24) / 2^44