Binomial Experiments: Conditions, Probabilities, and Summary Measures

All flashcards

What is a binomial experiment?

An experiment with a fixed number of independent trials, two possible outcomes per trial, and the same probability of success on every trial.

What do $p$ and $q$ represent?

$p$ is the probability of success on one trial, and $q$ is the probability of failure. They satisfy $p+q=1$ and $q=1-p$.

What does $X$ represent in a binomial setting?

$X$ is the number of successes obtained in the $n$ independent trials; its possible values are $0,1,\ldots,n$.

What is a Bernoulli trial?

A single trial with two possible outcomes and a fixed probability of success. A binomial experiment consists of one or more such trials under the binomial conditions.

Why can sampling without replacement violate the binomial conditions?

An earlier selection changes the pool of items remaining, which can change the probability of success on a later draw. The trials are then not independent with a constant success probability.

Quiz questions

1. If $X\sim B(18,0.3)$, what do the parameters indicate?Answer: There are 18 trials, with success probability 0.3 per trial.
2. For $X\sim B(n,p)$ with $q=1-p$, which expression gives the probability of exactly $x$ successes?Answer: $P(X=x)=\binom{n}{x}p^xq^{n-x}$
3. What probability does `binomcdf(n, p, k)` calculate for a binomial random variable $X$?Answer: $P(X\leq k)$
4. For a binomial random variable $X$, which expression gives the probability of at least $x$ successes?Answer: $1-P(X\leq x-1)$
5. For $X\sim B(n,p)$, what is the expected number of successes?Answer: $np$