> ## Documentation Index
> Fetch the complete documentation index at: https://v1-learn.neoartd.my.id/llms.txt
> Use this file to discover all available pages before exploring further.

# The Binomial Distribution

A `binomial experiment` is a probability
experiment that satisfies the following
four requirements:

> Eksperimen binomial adalah eksperimen probabilitas yang memenuhi empat persyaratan berikut:

1. Each trial can have only two outcomes
   or outcomes that can be reduced to two
   outcomes. Each outcome can be
   considered as either a success or
   a failure.

   > Setiap percobaan hanya memiliki dua hasil atau hasil yang dapat direduksi menjadi dua hasil. Setiap hasil dapat dianggap sebagai sukses atau kegagalan.

2. There must be a fixed number of trials.

   > Harus ada jumlah percobaan yang tetap.

3. The outcomes of each trial must be independent of each other.

   > Hasil dari setiap percobaan harus independen satu sama lain.

4. The probability of success must remain the same for each trial.
   > Probabilitas keberhasilan harus tetap sama untuk setiap percobaan.

The outcomes of a binomial
experiment and the corresponding
probabilities of these outcomes are
called a `binomial distribution.`

> Hasil dari eksperimen binomial dan probabilitas yang sesuai dari hasil-hasil ini disebut `distribusi binomial.`

<Card>
  Notation for the Binomial Distribution:

  * **$P(S) = p$**, probability of a success
  * **$P(F) = 1 - p = q$**, probability of a failure
  * **$n$** = number of trials
  * **$X$** = number of successes.
</Card>

## Binomial Probability Formula

<Card>
  In a binomial the probability of exactly $X$ successes in n trials is

  <br />

  $P(X) = \dfrac{n!}{(n-X)!X!} ~ p^X  q^{n-X}$
</Card>

### Binomial Probability - Example

<Card>
  If a student randomly guesses at five
  multiple-choice questions, find the
  probability that the student gets exactly
  three correct. Each question has five
  possible choices.

  <br />

  Solution:

  $n = 5, X = 3,\text{ and } p = \dfrac{1}{5}$. Then,

  <br />

  $P(3) = \dfrac{5!}{(5-3)!3!} \left( \dfrac{1}{5} \right)^3 \left( \dfrac{4}{5} \right)^2 \approx 0.05$
</Card>

<Card>
  A survey from Teenage Research
  Unlimited (Northbrook, Illinois.) found
  that 30% of teenage consumers
  received their spending money from
  part-time jobs. If five teenagers are
  selected at random, find the probability
  that at least three of them will have
  part-time jobs.

  <br />

  Solution:

  $n = 5, X = 3, 4, \text{ and } 5, \text{ and } p = 0.3$. Then,

  <br />

  $P(X \geq 3) = P(3) + P(4) + P(5) = 0.1323 + 0.0284 + 0.0024 = 0.1631$

  <Tip>
    You can use Table B in the textbook to find the Binomial probabilities as
    well.
  </Tip>
</Card>

<Card>
  A report from the Secretary of Health and Human Services stated that 70% of single-
  vehicle traffic fatalities that occur on weekend nights involve an intoxicated
  driver. If a sample of 15 single-vehicle
  traffic fatalities that occurred on a
  weekend night is selected, find the
  probability that exactly 12 involve a driver
  who is intoxicated.

  <br />

  Solution:

  $n = 15, X = 12, \text{ and } p = 0.70$. From Table B, $P(X = 12) = 0.170$
</Card>

<Card>
  A coin is tossed four times. Find the
  mean, variance, and standard deviation of
  the number of heads that will be obtained.

  <br />

  Solution:

  $n = 4, p = \dfrac{1}{2}, \text{ and } q = \dfrac{1}{2}$. Then,

  <br />

  $\mu = np = 4 \times \dfrac{1}{2} = 2$

  <br />$\sigma^2 = npq = 4 \times \dfrac{1} {2} \times \dfrac{1} {2} = 1$

  <br />

  $\sigma = \sqrt{1} = 1$
</Card>