An event has a probability p = \frac{5}{8}. Find

Step 1
P(0) Probability of exactly 0 successes
If using a calculator, you can enter ${\displaystyle \text{trials}=6,p=0.625,}$ and ${\displaystyle X=0}$ into a binomial probability distribution function (PDF). If doing this by hand, apply the binomial probability formula:
$$P(X)=(\begin{array}{c}n\\ x\end{array})\times p^x\times (1-p{)}^{n-x}$$
The binomial coefficient, $$(\begin{array}{c}n\\ x\end{array})$$ is defined by
$$(\begin{array}{c}n\\ x\end{array})=\frac{n!}{X!(n-X)!}$$
The full binomial probability formula with the binomial coefficient is
${\displaystyle P\left(X\right)=\frac{n!}{X!(n-X)!}\times p^x\times {(1-p)}^{n-x}}$
Where n is the number of trials, p is the probability if success on a single trial, and X is the number of successes. Substituting in values for this problem, ${\displaystyle n=6,p=0.625,}$ and ${\displaystyle X=0}$
${\displaystyle P\left(0\right)=\frac{6!}{0!(6-0)!}\times {0.625}^0\times {(1-0.625)}^{6-0}}$
Evaluting the expression, we have
${\displaystyle P\left(0\right)=0.0027809143066406}$
Step 2
If we apply the binomial probability formula, or a calculator's binomial probability distribution (PDF) function, to all possible values of X for 6 trials, we can construct a complete binomial distribution table. The sum of the probabilities in this table will always be 1. The complete binomial distribution table for this problem, with ${\displaystyle p=0.625}$ and 6 trials is:
${\displaystyle P\left(0\right)=0.0027809143066406}$
${\displaystyle P\left(1\right)=0.027809143066406}$
${\displaystyle P\left(2\right)=0.11587142944336}$
${\displaystyle P\left(3\right)=0.25749206542969}$
${\displaystyle P\left(4\right)=0.32186508178711}$
${\displaystyle P\left(5\right)=0.21457672119141}$
${\displaystyle P\left(6\right)=0.059604644775391}$