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Binomial function how do i use the binomial function on my calculator? (what do I enter in order to solve a problem)?

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Hello,

The binomial probability distribution is defined as
P(r;p;n) =(nCr)(p^r)*(1-p)^(n-r)
where n is the number of trials, p the probability of success, and r the expected result.

Let n=20, r=7, p=0.15 ( I do not know wether this exemple has any meaning in the context of your problem, but you have to enter values that mean something to you. I am only showing you the key strokes

To enter 20C7 you press 20 [SHIFT][nCr]7 ;
To enter 0.15 to the power 7 you type 0.15[X to ] 7 the key is between
[x²] and [log]

To enter (1-0.15) to power 20-7, you type 0.85 [X to] 13
With [*] standing for multiplication key , and [X to] the raise to power key, the exemple above can be entered as

( 20 [SHIFT][nCr] 7) [*] ( 0.15 [X to] 7 ) [*] ( 0.85 [X to] 13 ) [=]

Here is a screen capture to show you what it looks like. However on this calculator the combination 20 [SHIFT][nCr] 7 is represented as nCr(20,7).

binomial function - e74be60.jpg

Hope it helps

Posted on Nov 15, 2009

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TI-89 Titanium: I want to solve a Binomial Theorem problem (x+y)^6 how would i go about solving this in the calculator?


Using elementary algebria in the binomial theorem, I expanded the power (x + y)^n into a sum involving terms in the form a x^b y^c. The coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. This is known as binomial coefficients and are none other than combinatorial numbers.

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Using binomial coefficient (n over k) allowed me to choose k elements from an n-element set. This you will see in my calculations on my Ti 89. This also allowed me to use (x+y)^n to rewrite as a product. Then I was able to combine like terms to solve for the solution as shown below.
(x+y)^6= (x+y)(x+y)(x+y)(x+y)(x+y)(x+y) = x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6

This also follows Newton's generalized binomial theorem:


oneplusgh_15.jpg
Now to solve using the Ti 89.


Using sigma notation, and factorials for the combinatorial numbers, here is the binomial theorem:

oneplusgh.gif

The summation sign is the general term. Each term in the sum will look like that as you will see on my calculator display. Tthe first term having k = 0; then k = 1, k = 2, and so on, up to k = n.
Notice that the sum of the exponents (n ? k) + k, always equals n.



oneplusgh_26.jpg


The summation being preformed on the Ti 89. The actual summation was preformed earlier. I just wanted to show the symbolic value of (n) in both calculations. All I need to do is drop the summation sign to the actual calculation and, fill in the term value (k), for each binomial coefficient.



oneplusgh_18.jpg

This is the zero th term. x^6, when k=0. Notice how easy the calculations will be. All I'm doing is adding 1 to the value of k.


oneplusgh_19.jpg

This is the first term or, first coefficient 6*x^5*y, when k=1.
Solution so far = x^6+6*x^5*y



oneplusgh_20.jpg


This is the 2nd term or, 2nd coefficient 15*x^4*y^2, when k=2.
Solution so far = x^6+6*x^5*y+15*x^4*y^2



oneplusgh_27.jpg



This is the 3rd term or, 3rd coefficient 20*x^3*y^3, when k=3.
Solution so far = x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3



oneplusgh_28.jpg



This is the 4th term or, 4th coefficient 15*x^2*y^4, when k=4.
Solution so far = x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4



oneplusgh_21.jpg


This is the 5th term or, 5th coefficient 6*x*y^5, when k=5.
Solution so far = x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5



oneplusgh_22.jpg

This is the 6th term or, 6th coefficient y^6, when k=6.
Solution so far = x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5+y^6



oneplusgh_23.jpg

Putting the coefficients together was equal or, the same as for when I used the expand command on the Ti 89.

binomial coefficient (n over k) for (x+y)^6
x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5+y^6












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1 Answer

When i enter the binomial probability formula into the casio fx-115es it will not operate. it says that the problem lies in the (n-X)!X! part of the formula. I'm writing everything exactly the way it is in...


You may be writing the expression as they appear in the formula, but the problem stems from the fact the factorial function increase rather rapidly and you cannot calculate the factorials of numbers larger that 69 which is the limit of the calculator.
However if you use the built in Combination function nCr, you will avoid the problem. In the binomial function, the n!/(r!(n-r)!) factor can be replaced by nCr or nC(n-r).

Do not use the explicit form with the factorials because you will get an overflow.

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Hello,

The binomial probability distribution is defined as
P(r;p;n) =(nCr)(p^r)*(1-p)^(n-r)
where n is the number of trials, p the probability of success, and r the expected result.

Let n=20, r=7, p=0.15 ( I do not know wether this exemple has any meaning in the context of your problem, but you have to enter values that mean something to you. I am only showing you the key strokes

To enter 20C7 you press 20 [SHIFT][nCr]7 ;
To enter 0.15 to the power 7 you type 0.15[X to ] 7 the key is between
[x²] and [log]

To enter (1-0.15) to power 20-7, you type 0.85 [X to] 13
With [*] standing for multiplication key , and [X to] the raise to power key, the exemple above can be entered as

( 20 [SHIFT][nCr] 7) [*] ( 0.15 [X to] 7 ) [*] ( 0.85 [X to] 13 ) [=]

Here is a screen capture to show you what it looks like. However on this calculator the combination 20 [SHIFT][nCr] 7 is represented as nCr(20,7).

e74be60.jpg

Hope it helps

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