When we flip a coin once there are 2^{1}=2 possible outcomes – H (heads) or T (tails). When a coin is tossed two times there are 2^{2}=4 possible outcomes – HH, HT, TH, TT. Similarly, when we flip a coin 100 times the total number of possible outcomes is 2^{100}= 1267650600228229401496703205376. Note that it is not possible to explicitly enumerate the total number of possible outcomes since the number is so large.

**Calculating Probabilities of getting Heads when flipping a coin 100 times:**

Let X denote the number of heads that we obtain when the coin is tossed n=100 times. The probability of getting heads in a single coin toss is equal to p = 1/2. Similarly, the probability of getting tails in a single coin toss is equal to q=1/2. We can calculate the probability of getting ‘x’ number of heads by using the formula for the binomial distribution. The required probability is given as, P(X=x) = {100 \choose x} p^x q^{100-x} = \frac{100!}{x!(100-x)!}\left(\frac{1}{2}\right)^x \left(\frac{1}{2}\right)^{100-x}.

We can also sometimes calculate the probabilities by relying upon symmetry consideration instead of the above formula. the disadvantage of the above formula is that the calculations can sometimes be too large and unwieldy. In such cases, symmetry considerations give us a simpler and direct answer. For example, we can say that the probability of getting more than 50 heads is approximately equal to 1/2. This is because by symmetry the probabilities of getting more than 50 heads should be equal to the probability of getting more than 50 tails (which is the same as getting less than 50 heads). Since these are all the possible outcomes (neglecting the case when we get exactly 50 heads) both of them should have a probability roughly equal to 1/2. We say roughly 1/2 and not exactly 1/2 because we have neglected one possible outcome when we get* exactly *50 heads.

**Expected Value, Variance and Standard Deviation:**

We can calculate the expected value of the number of heads using the formula, E(X) = np = 100 \times 1/2 = 50. So we can expect to get around 50 heads. The variance is given by the formula, V(X) = npq = 100 \times 1/2 \times 1/2 = 25. Hence, the standard deviation which is the square root of the variance is equal to 5.

The empirical rule tells us that there is a 99.7% chance that the number of heads lies within three standard deviations of the expected value. Therefore, there is a 99.7% probability that the number of heads lies between 35 and 65.