binomial distribution

(noun)

The discrete probability distribution of the number of successes in a sequence of $n$ independent yes/no experiments, each of which yields success with probability $p$.

Related Terms

Examples of binomial distribution in the following topics:

  • Experimental Probabilities

    • In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modeling using a binomial distribution might be appropriate.
    • A binomial distribution is the discrete probability distribution of the number of successes in a sequence of $n$ independent yes/no experiments.
    • A model-based alternative would be to select of family of probability distributions and fit it to the data set containing the values of years past.
    • The fitted distribution would provide an alternative estimate of the desired probability.

  • Binomial Expansions and Pascal's Triangle

    • The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial.
    • The binomial theorem is an algebraic method of expanding a binomial expression.
    • This formula is referred to as the Binomial Formula.
    • Applying these numbers to the binomial expansion, we have:
    • Use the Binomial Formula and Pascal's Triangle to expand a binomial raised to a power and find the coefficients of a binomial expansion

  • Complex Numbers and the Binomial Theorem

    • Powers of complex numbers can be computed with the the help of the binomial theorem.
    • Recall the binomial theorem, which tells how to compute powers of a binomial like $x+y$.
    • Using the binomial theorem directly, this can be written as:
    • Recall that the binomial coefficients (from the 5th row of Pascal's triangle) are $1, 5, 10, 10, 5, \text{and}\, 1.$ Using the binomial theorem directly, we have:
    • Connect complex numbers raised to a power to the binomial theorem

  • Multiplying Algebraic Expressions

    • Multiplying two binomials is less straightforward; however, there is a method that makes the process fairly convenient.
    • "FOIL" is a mnemonic for the standard method of multiplying two binomials (hence the method is often referred to as the FOIL method).
    • Outer (the "outside" terms are multiplied—i.e., the first term of the first binomial with the second term of the second)
    • Inner (the "inside" terms are multiplied—i.e., the second term of the first binomial with the first term of the second)
    • Remember that any negative sign on a term in a binomial should also be included in the multiplication of that term.

  • Binomial Expansion and Factorial Notation

    • The binomial theorem describes the algebraic expansion of powers of a binomial.
    • Recall that the binomial theorem is an algebraic method of expanding a binomial that is raised to a certain power, such as $(4x+y)^7$.
    • The coefficients that appear in the binomial expansion are called binomial coefficients.
    • Example: Use the binomial formula to find the expansion of $(x+y)^4$
    • Use factorial notation to find the coefficients of a binomial expansion

  • Addition, Subtraction, and Multiplication

    • $(a + bi)(c + di) = ac + bci + adi + bidi$ (by the distributive law)
    • Discover the similarities between arithmetic operations on complex numbers and binomials

  • Total Number of Subsets

    • The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.
    • The binomial theorem describes the algebraic expansion of powers of a binomial.
    • The coefficient a in the term of $ax^by^c$ is known as the binomial coefficient $n^b$ or $n^c$ (the two have the same value).
    • Employ the Binomial Theorem to find the total number of subsets that can be made from n elements

  • Sums, Differences, Products, and Quotients

    • A monomial equations has one term; a binomial has two terms; a trinomial has three terms.
    • Multiplying binomials and trinomials is more complicated, and follows the FOIL method.
    • FOIL is a mnemonic for the standard method of multiplying two binomials; the method may be referred to as the FOIL method.
    • Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
    • Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)

  • Finding a Specific Term

    • The rth term of the binomial expansion can be found with the equation: ${ \begin{pmatrix} n \\ r-1 \end{pmatrix} }{ a }^{ n-(r-1) }{ b }^{ r-1 }$.
    • You might multiply each binomial out to identify the coefficients, or you might use Pascal's triangle.
    • Let's go through a few expansions of binomials, in order to consider any patterns that are present in the terms.

  • Factoring Perfect Square Trinomials

    • When a trinomial is a perfect square, it can be factored into two equal binomials.
    • Note that if a binomial of the form $a+b$ is squared, the result has the following form: $(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2.$ So both the first and last term are squares, and the middle term has factors of $2, $ $a$, and $b,$ where the latter are the square roots of the first and last term respectively.