Lesson Binomial Experiments

Source code of 'Binomial Experiments'

This Lesson (Binomial Experiments) was created by by Nate(3500)

*Caution* To understand this lesson be sure to you know: factorial, combinations, and basic understandings of probabilities.
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Binomial experiments are easy ways to determine the possible outcome of a situation. By the prefix 'bi', we know these experiments deal with items in twos.
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Form: (T+F)^0 = {{{1T^0F^0}}}
The power (zero) determines the amount of terms that result. Do this by adding one to the exponent, so (T+F)^0 has only one term.
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Form: (T+F)^1
We know that there should be two terms resulting. {{{(T+F)^1=T^1F^0+T^0F^1}}}
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Form: (T+F)^2 
This should have three terms.
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Now, let us determine how to find the coefficients of the terms. This is easy to find.
You need to use the exponent.
Let us use (T+F)^4 ....
To find the coefficient of the first term (T^4F^0), use combinations. C(4,4) is the coefficient of the first term. The coefficient is 1. For the second term (T^3F^1), use C(4,3). The coefficient is 4. For the third term (T^2F^2), use C(4,2). The coefficient is 6. After using combinations more and more, you can make the form....
{{{(T+F)^4=T^4+4T^3F+6T^2F^2+4TF^3+F^4}}}
Notice that the first and last coefficient are the same. Also, notice the second from last and the second coefficient are the same. This is a general rule that is always true.
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If you haven't noticed, to determine the variables to use is easy:
For (T+F)^3, use descending order. The variables used would be: {{{T^3F^0+T^2F^1+T^1F^2+T^0F^3}}}
Notice that the sum of the exponents of the two variables is equal to the exponent of (T+F)^3
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Now, lets use binomial experiments to determine probability:
You flip a coin five time; what is the probility of getting all Heads?
{{{(H+T)^5 = _H^5T^0+_H^4T^1+_H^3T^2+_H^2T^3+_H^1T^4+_H^0T^5}}}
C(5,5)=1
C(5,4)=5
C(5,3)=10
C(5,2)=10
C(5,1)=5
C(5,0)=1
{{{(H+T)^5 = 1H^5T^0+5H^4T^1+10H^3T^2+10H^2T^3+5H^1T^4+1H^0T^5}}}
The H is the amount of heads while the T is the amount of tails. The exponent is used to determine the amount of heads while the coefficient tells us how many times that possible outcome can occur.
What is the probility of getting all Heads?
To answer this question, look at the term 1H^5T^0. In this term, there are five heads and zero tails. The possible ways to achieve this is 1 due to the coefficient. Now, take the sum of all the coefficients. You should get 32, so the probability of getting 5 heads is 1/32.
What is the probability of getting atleast 3 tails?
Look at the terms 10H^2T^3, 5H^1T^4, and 1H^0T^5. These terms tell you that the amount of tails is atleast three. Take the sum of all the coefficients of these terms. You should get 16. The probability is 16/32 or 1/2.
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Now, I am going to use the same equation as above, but H is the success at a game machine while T is the failing. You play Five Times!
Example: The chance of winning on the game is 0.25; what is chance of --->
Winning All Five Times: 1H^5T^0 = (1/4)^5 = 1/1024 = 0.0977%
Losing All Five Times: 1H^0T^5 = (3/4)^5 = 243/1024 = 23.7% *WOW, low chance of losing all five times!
Winning Atleast Four Times: 1H^5T^0 + 5H^4T^1 = (1/4)^5 + 5(1/4)^4(3/4) = 1/1024 + 15/1024 = 16/1024 = 1/64 = 1.5625%