Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

10. Bernoulli Processes

  • Page ID
    24957
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    The following topics are included in the seven videos below.

    1. Intro to Bernoulli Processes
    2. Bernoulli Processes, Example 1 and definitions
    3. Bernoulli Processes, Example 2
    4. Bernoulli Processes, Example 3
    5. Bernoulli Processes, Example 4 (with conditional)
    6. Bernoulli Processes, Example 5 (using a complement)
    7. Bernoulli Processes, Example 6 (with conditional)

     

     

     

     

     

     

     

     

    Prework

    1. A fair die is tossed five times. Find the probability of throwing four sixes.

    2. A fair die is tossed five times. Find the probability of throwing at least 1 six.

    3. The patrons at a certain library check out only fiction books 70% of the time. Assume that the visits to the library are independent. What is the probability that 20 of the next 25 patrons will check out only fiction books?

    Google Form

    Solutions

    1. A success is throwing a six while a failure is throwing anything else. On one toss, the probability of throwing a six is \(\frac{1}{6}\) so \(p=\frac{1}{6}\) and therefore \(1-p=\frac{5}{6}\). We roll the die \(5\) times so \(n=5\) and we want four sixes, so \(r=4\). Therefore, Pr(four sixes)\(=C(5,4)\cdot (\frac{1}{6})^4\cdot (\frac{5}{6})^1\).
    2. Everything is the same as in problem 1, except that \(r=1,2,3,4,5\). We could plug each of these into the formula and add them together, or we we can use the complement. In the complement, \(r=0\). Therefore Pr(at least 1 six)\(=1-C(5,0)\cdot (\frac{1}{6})^0\cdot (\frac{5}{6})^5\).
    3. A success is a library patron checking out only fiction while a failure is a library patron not checking out only fiction. We also see that \(p=0.7, 1-p=0.3, n=25\) and \(r=20\). So Pr(20 check out only fiction)\(=C(25,20)\cdot (0.7)^{20}\cdot (0.3)^5\).


    10. Bernoulli Processes is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?