a perso increases the number of tosses of a fair coin the actual number of heads will get farther and farther away from the number of tosses divided by 2

It will fluctuate about an equal number of heads and tails.

The statement is false.

To understand why the number of heads will get farther away from the number of tosses divided by 2 as the number of tosses increases, we need to examine the concept of probability and the law of large numbers.

The probability of getting a head on any given coin toss is 1/2 since the coin is fair. This means that if we toss the coin many times, we would expect approximately half of the tosses to result in heads. However, this is not guaranteed for every set of tosses.

When we perform a small number of coin tosses, the ratio of heads to tosses can deviate significantly from 1/2 due to random variation. For example, if we flip a coin 10 times, we might get 6 heads and 4 tails. In this case, the ratio is 6/10 or 0.6, which is different from 1/2.

However, as we increase the number of coin tosses, the law of large numbers tells us that the observed proportion of heads will converge to the true probability of getting a head, which is 1/2 in the case of a fair coin.

As an example, if we flip the coin 100 times, we would expect to get around 50 heads. However, the actual number of heads might be slightly different, such as 52 or 48. The difference between the actual number of heads and 50 will become smaller as we increase the number of tosses.

To understand why the difference between the actual number of heads and the number of tosses divided by 2 becomes smaller with more tosses, we can consider the concept of standard deviation and the binomial distribution. The standard deviation of a binomial distribution, which models the number of heads in a series of coin tosses, is given by the formula sqrt(npq), where n is the number of tosses, p is the probability of getting a head, and q is the probability of getting a tail.

As we increase the number of tosses (n), the standard deviation becomes larger, indicating that the observed number of heads can deviate more from the expected value of n/2. However, due to the law of large numbers, the ratio of heads to tosses will still converge to 1/2 on average, and the difference from n/2 will become smaller relative to the larger number of tosses.

In summary, the more coin tosses we perform, the closer the observed ratio of heads to tosses will be to 1/2, the true probability of getting a head. However, there can still be significant deviations in the actual number of heads for a given number of tosses due to random variation, especially with a small number of tosses.