Bernoulli Distribution:

This distribution best describes all situations where a "trial" is made resulting in either "success" or "failure," such as when tossing a coin, or when modeling the success or failure of a surgical procedure. The Bernoulli distribution is defined as:
f(x) = px (1-p)1-x,    for x = 0, 1,
where
p is the probability that a particular event (e.g., success) will occur.

Binomial Distribution

Suppose we repeat a Bernouilli p experiment n times and count the number X of successes, the distribution of X is called the Binomial B(n,p) random variable.

Probability mass function:

\begin{displaymath}P(X=k)={\binom{n}{k}} p^kq^{(n-k)}\end{displaymath},

where q = 1 - p,    k=0, 1, 2, ..., n.

E(X) = np

Var(X) = np(1-p)

Odds Ratios and Mode: The odds of k successes relative to (k-1) successes are:

\begin{displaymath}\frac{P(X=k)}{P(X=k-1)}=\frac{n-k+1}{k}\frac{p}{q}\end{displaymath}

This is very useful for computing by recursion the probability mass of the binomial.

Property:

For X a B(n,p) random variable with probability of success p neither 0 or 1, then as k varies from 0 to n, P(X=k) first increases monotonically and then decreases monotonically, (it is unimodal) reaching its highest value when k is the largest integer less or equal to (n+1)p.

Proof:

\begin{displaymath}P(X=k) \geq P(X=k-1)\end{displaymath}

is equivalent to
\begin{displaymath}(n-k+1)p\geq k(1-p)\; iff\; (n+1)p \geq k\end{displaymath}

The value where the the probability mass function takes on its maximum is called the mode.

Examples:

  1. A manufacturer of nails claims that only 3% of its nails are defective. a random sample of 24 nails is selected, and it is found that two of them are defective.  Is it fair to reject the manufacturer's claim based on this observation.

  2. A certain rare blood type can be found in only  0.05% of people.  If the population of a randomly selected group is 3000, what is the probability that at least two persons in the group have this rare blood type.