Define probability distribution function of binomial distribution. Assume that the probability of
a boat reaching its destination safely is 0.9. Under the assumption of binomial distribution, find
the mean and standard deviation of boats reaching destination safely out of a total 500 boats.
A probability distribution function (PDF) or probability mass function (PMF) of a binomial distribution describes the likelihood of obtaining a specific number of successes (usually denoted as "k") in a fixed number of independent Bernoulli trials. In a binomial distribution, each trial has only two possible outcomes: success or failure, and the probability of success is denoted as "p." The binomial distribution is characterized by two parameters: "n," the number of trials, and "p," the probability of success on each trial.
The probability mass function of a binomial distribution is given by:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes.
- C(n, k) represents the binomial coefficient, which is equal to (n choose k), calculated as C(n, k) = n! / (k! * (n - k)!), where "!" denotes the factorial.
- p is the probability of success in a single trial.
- (1 - p) is the probability of failure in a single trial.
- n is the total number of trials.
- k
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