Bernoulli trials and binomial probability

Bernoulli trials with two outcomes, binomial experiments with repeated independent trials, the binomial probability formula P(X = k) = nCk p^k q^(n−k), and probabilities of at least one and at most one success.

Bernoulli trials

A Bernoulli trial has exactly two outcomes, called success and failure, with $P(\text{success}) = p$ and $P(\text{failure}) = q = 1 - p$. Tossing a coin and checking whether a die shows a six are both Bernoulli trials.

Binomial experiments

A binomial experiment repeats a Bernoulli trial $n$ times, where the trials are independent and $p$ stays the same. Let $X$ be the number of successes.

Binomial probability

$$P(X = k) = \binom{n}{k} p^k q^{\,n-k}, \qquad k = 0, 1, 2, \dots, n$$

The $p^k q^{n-k}$ is the probability of one particular order of $k$ successes and $n - k$ failures, and $\binom{n}{k}$ counts the orders.

Example 1●●

A die is rolled $5$ times. Find the probability of exactly two sixes.

$n = 5$, $p = \tfrac16$, $q = \tfrac56$.

$P(X = 2) = \dbinom{5}{2}\left(\tfrac16\right)^2\left(\tfrac56\right)^3 = 10 \times \dfrac{125}{7776} \approx 0.1608$.

Example 2: at least one●●

A die is rolled $4$ times. Find the probability of at least one six.

Use the complement: $P(X \ge 1) = 1 - P(X = 0) = 1 - \left(\tfrac56\right)^4 = \dfrac{671}{1296} \approx 0.5177$.

Flashcards

Bernoulli trial

One trial with two outcomes: success with probability $p$, failure with $q = 1 - p$.

Binomial conditions

$n$ independent trials, each with the same $p$.

$P(X = k)$

$\dbinom{n}{k}p^k q^{n-k}$

$P(X ge 1)$

$1 - q^n$

Practice

A coin is tossed $4$ times. Find the probability of exactly $2$ heads.
A die is rolled $5$ times. Find the probability of exactly two sixes, correct to four decimal places.
A die is rolled $4$ times. Find the probability of at least one six, correct to four decimal places.
$p = 0.3$ and there are $3$ trials. Find the probability that all three are successes.
$p = 0.2$ and there are $5$ trials. Find the probability of no successes, correct to four decimal places.
In how many orders can $2$ successes occur in $5$ trials?
Which is a Bernoulli trial?
Drawing $3$ cards from a deck without replacement and counting hearts is not binomial because: