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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.

Notation for the Binomial Distribution:
  • P(S)=pP(S) = p, probability of a success
  • P(F)=1−p=qP(F) = 1 - p = q, probability of a failure
  • nn = number of trials
  • XX = number of successes.

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Binomial Probability Formula

In a binomial the probability of exactly XX successes in n trials is
P(X)=n!(n−X)!X! pXqn−XP(X) = \dfrac{n!}{(n-X)!X!} ~ p^X q^{n-X}

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Binomial Probability - Example

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.
Solution:n=5,X=3, and p=15n = 5, X = 3,\text{ and } p = \dfrac{1}{5}. Then,
P(3)=5!(5−3)!3!(15)3(45)2≈0.05P(3) = \dfrac{5!}{(5-3)!3!} \left( \dfrac{1}{5} \right)^3 \left( \dfrac{4}{5} \right)^2 \approx 0.05

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.
Solution:n=5,X=3,4, and 5, and p=0.3n = 5, X = 3, 4, \text{ and } 5, \text{ and } p = 0.3. Then,
P(X≥3)=P(3)+P(4)+P(5)=0.1323+0.0284+0.0024=0.1631P(X \geq 3) = P(3) + P(4) + P(5) = 0.1323 + 0.0284 + 0.0024 = 0.1631
You can use Table B in the textbook to find the Binomial probabilities as well.

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.
Solution:n=15,X=12, and p=0.70n = 15, X = 12, \text{ and } p = 0.70. From Table B, P(X=12)=0.170P(X = 12) = 0.170

A coin is tossed four times. Find the mean, variance, and standard deviation of the number of heads that will be obtained.
Solution:n=4,p=12, and q=12n = 4, p = \dfrac{1}{2}, \text{ and } q = \dfrac{1}{2}. Then,
μ=np=4×12=2\mu = np = 4 \times \dfrac{1}{2} = 2
σ2=npq=4×12×12=1\sigma^2 = npq = 4 \times \dfrac{1} {2} \times \dfrac{1} {2} = 1
σ=1=1\sigma = \sqrt{1} = 1