Coin Flipper Calculator

Each coin flip is simulated using a hash-based pseudo-random number generator. For the $$i$$-th flip, the calculator computes a hash value $$h_i$$ using sinusoidal hashing:

$$h_i = |\sin(\text{seed} \times a_i + b_i) \times c_i| \mod 1$$

where $$a_i$$, $$b_i$$, and $$c_i$$ are unique constants for each flip position, producing values uniformly distributed between 0 and 1. The outcome rule is:

$$\text{Result}_i = \begin{cases} \text{Heads} & \text{if } h_i > 0.5 \\ \text{Tails} & \text{if } h_i \leq 0.5 \end{cases}$$

The heads count is the sum of all heads outcomes:

$$\text{Heads Count} = \sum_{i=1}^{n} \mathbf{1}(h_i > 0.5)$$

The heads percentage is calculated as:

$$\text{Heads \%} = \frac{\text{Heads Count}}{n} \times 100$$

For a fair coin, the expected value of heads in $$n$$ flips follows a binomial distribution $$B(n, 0.5)$$ with expected value $$E[X] = n/2$$ and standard deviation $$\sigma = \sqrt{n \times 0.5 \times 0.5}$$.

With a fair coin, you expect approximately 50% heads over many flips. However, small sample sizes (1-10 flips) naturally show significant deviation from 50%. This is known as sampling variability. For example, getting 7 heads out of 10 flips (70%) is not unusual and has a probability of about 11.7% with a fair coin. The law of large numbers states that as the number of flips increases, the observed proportion converges toward the theoretical probability of 0.5.

Different seed values produce different outcome sequences, allowing you to explore the range of possible results. If you observe a particular seed consistently producing extreme results, remember this is a property of the pseudo-random sequence, not evidence of bias.