Probability distributions are a way of describing all possible events and probability of each one happening. Probability distributions are useful for asking questions about ranges of possible values.
Binomial distribution is used to calculate the probability of a certain number of successful outcones, and it involves three parameters:
Probability Mass Function (pmf) for binomial distribution:
$$\text{B}(k; n, p) = \binom{n}{k} \times p^k \times (1 - p)^{n - k}$$
Beta distribution is used to estimate the probability of an event for which we have already observed a number of trials and the number of successful outcomes. For example, we would use it to estimate the probability of heads, when so far we have observed 100 tosses of a coin and 40 of those were heads.
Probability density function of beta distribution:
$$\text{Beta}(p, \alpha, \beta) = \frac{ p^{\alpha - 1} \times (1 - p)^{\beta-1} }{ \text{beta}(\alpha, \beta) }$$