Lesson Challenging problems on Binomial distribution probability

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Challenging problems on Binomial distribution probability


Problem 1

Let  X  represents the difference between the number of heads and the number of tails
when a coin is tossed  33  times.  Find P (X=3).

Solution

In this problem, the sum  Heads+Tails  is 33,  while the difference Heads-Tails = 3.


It immediately implies that Heads = 18, Tails = 15.


Therefore, the event X= 3 is EQUIVALENT to event Tails = 15.


Now, P(Tails=15) is (use the formula of probability for the Binomial distribution)


    P(Tails=15) = C%5B33%5D%5E15%2A%281%2F2%29%5E15%2A%281%2F2%29%5E18 = BIMON.DIST(15, 33, 0.5, FALSE) = 0.120741.    ANSWER


I used the standard EXCEL function to calculate the binomial distribution probability.


        Its first  parameter is the "number of successful trials"   (k= 15 in this case).
        Its second parameter is the "number of trials n"           (n= 33 in this case).
        Its third  parameter is "the probability of success in each one single trial"  (p= 0.5 in this case); and 
        The fourth parameter says if you want calculate a single addend or the sum of addends (the single addend in this case).

    Input  "=BINOM.DIST(15, 33, 0.5, FALSE)" into any MS Excel cell in your computer and press enter to get the answer P = 0.120741 immediately.


    On Excel function BINOM.DIST, see its description everywhere, for example

    https://support.office.com/en-us/article/binom-dist-function-c5ae37b6-f39c-4be2-94c2-509a1480770c

Problem 2

A person flips a coin to determine whether she's moving forwards  or backward.
If she flips a head,  she moves  1  step forward,  and if she flips a tail,  she moves  1 step  backward.
What is the probability that after  4  coin tosses,  she will be in the same place?

Solution

It is the probability of getting two tails and two heads of 4 flips in any order


    P(2T2H of 4) = C%5B4%5D%5E2%2A%281%2F2%29%5E2%2A%281%2F2%29%5E2 = %28%284%2A3%29%2F%281%2A2%29%29%2A%281%2F2%29%5E4 = 0.375.     ANSWER


My other Additional lessons on Probability in this site are
    - Sample space conception problems REVISITED
    - Solving probability problems using complementary probability REVISITED
    - Elementary Probability problems related to combinations REVISITED
    - Conditional probability problems REVISITED
    - More problems on Conditional probability
    - Dependent and independent events REVISITED
    - Elementary operations on sets help solving Probability problems - REVISITED

    - Simple and simplest probability problems on Binomial distribution
    - Typical binomial distribution probability problems
    - How to calculate Binomial probabilities with Technology (using MS Excel)
    - Solving problems on Binomial distribution with Technology (using MS Excel)
    - Solving problems on Binomial distribution with Technology (using online solver)

    - Using general probability formulas for a union or intersection of events
    - Twisted probability problems on intersections and unions of sets of events
    - Miscellaneous problems on Probability
    - The chances to be rescued from an inhabitant island
    - Analyzing chains of random events

    - Math expectation of winning in games problems
    - Math expectation of winning in lottery problems
    - Math expectation of winning in games with rolling pair of dice

    - Problems on uniformly distributed random variables
    - Classic problems of Elementary Geometric Probability theory
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    - OVERVIEW of my additional lessons on Probability

Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.


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