The difference between the number of heads and tails (represented by x) when a fair coin is flipped 49 times can range from -49 to 49, with 0 being the most likely outcome. This is because a fair coin has an equal chance of landing heads or tails on each flip.
To find the probability of a particular outcome, we can use the binomial distribution. The binomial distribution gives the probability of getting exactly k heads in n coin flips of a fair coin, where k can range from 0 to n. In this case, n = 49 and k can range from 0 to 49.
The formula for the binomial distribution is given by:
P(k heads in n flips) = (n choose k) * (0.5)^k * (0.5)^(n-k)
where (n choose k) is the number of ways to choose k heads from n flips, given by the binomial coefficient:
(n choose k) = n! / (k! * (n-k)!).
So, the probability of getting x heads in 49 flips, where x = k - (49 - k) = 2k - 49, can be calculated as:
P(x heads in 49 flips) = (49 choose k) * (0.5)^k * (0.5)^(49-k)
where k = (49 + x) / 2.
Note that the binomial distribution is symmetric around the mean, which in this case is 49 / 2 = 24.5 heads. This means that the probabilities for getting x heads and getting -x heads will be the same.