Maintenance script: Automated superscript correction

2016-02-24T22:24:23Z

Automated superscript correction

← Older revision		Revision as of 22:24, 24 February 2016
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	# cumSum(binomcdf(N,0 gives the list {1 2 ... N+1}, and cumSum(not(binompdf(N,0 gives the list {0 1 2 ... N}.		# cumSum(binomcdf(N,0 gives the list {1 2 ... N+1}, and cumSum(not(binompdf(N,0 gives the list {0 1 2 ... N}.
	# With seq(, you normally do math inside the list: for example, seq(3I^^2^^,I,0,5		# With seq(, you normally do math inside the list: for example, seq(3I<sup>2</sup>,I,0,5
	# With these commands, you do the same math outside the list: 3Ans^^2^^ where Ans is the list {0 1 ... 5}.		# With these commands, you do the same math outside the list: 3Ans<sup>2</sup> where Ans is the list {0 1 ... 5}.

	An example:		An example:

KermMartian at 19:20, 24 February 2016

2016-02-24T19:20:55Z

← Older revision		Revision as of 19:20, 24 February 2016
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	The value of binompdf( is given by the formula		The value of binompdf( is given by the formula

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\operatorname{binompdf}(n,p,k) = \binom{n}{k}\,p^k\,(1-p)^{n-k} = \frac{n!}{k!\,(n-k)!}\,p^k\,(1-p)^{n-k}		\operatorname{binompdf}(n,p,k) = \binom{n}{k}\,p^k\,(1-p)^{n-k} = \frac{n!}{k!\,(n-k)!}\,p^k\,(1-p)^{n-k}
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	This formula is fairly intuitive. We want to know the probability that out of n trials, exactly k will be successes, so we take the probability of k successes - <math>p^k</math> - multiplied by the probability of (n-k) failures - <math>(1-p)^{n-k}</math> - multiplied by the number of ways to choose which k trials will be successes - <math>\binom{n}{k}</math>.		This formula is fairly intuitive. We want to know the probability that out of n trials, exactly k will be successes, so we take the probability of k successes - <math>p^k</math> - multiplied by the probability of (n-k) failures - <math>(1-p)^{n-k}</math> - multiplied by the number of ways to choose which k trials will be successes - <math>\binom{n}{k}</math>.

Maintenance script: Initial automated import

2016-02-24T18:23:53Z

Initial automated import

New page

{{Template:TI-BASIC:Command
|picture=BINOMPDF.GIF
|summary=Calculates the binomial probability, either at a single value or for all values
|syntax=for a single value:
binompdf(''trials'', ''probability'', ''value''

for a list of all values (0 to ''trials'')
binompdf(''trials'', ''probability''
|location=Press:
# 2ND DISTR to access the distribution menu
# 0 to select binompdf(, or use arrows.
Press ALPHA A instead of 0 on a TI-84+/SE with OS 2.30 or higher.
|compatibility=TI-83/84/+/SE
|size=2 bytes
}}

This command is used to calculate the binomial probability. In plainer language, it solves a specific type of often-encountered probability problem, that occurs under the following conditions:
# A specific event has only two outcomes, which we will call "success" and "failure"
# This event is going to repeat a specific number of times, or "trials"
# Success or failure is determined randomly with the same probability of success each time the event occurs
# We're interested in the probability that there are exactly N successes

For example, consider a couple that intends to have 4 children. What is the probability that 3 of them are girls?

# The event here is a child being born. It has two outcomes "boy" or "girl". We can call either one a success, but we'll choose to be sexist towards guys and call a girl a success in this problem
# The event is going to repeat 4 times, so we have 4 trials
# The probability of a girl being born is 50% or 1/2 each time
# We're interested in the probability that there are exactly 3 successes (3 girls)

The syntax here is binompdf(''trials'', ''probability'', ''value''). In this case:

:binompdf(4,.5,3

This will give .25 when you run it, so there's a .25 (1/4) probability out of 4 children, 3 will be girls.

An alternate syntax for binompdf( leaves off the last argument, ''value''. This tells the calculator to compute a list of the results for all values. For example:

:binompdf(4,.5

This will come to {.0625 .25 .375 .25 .0625} when you run it. These are the probabilities of all 5 outcomes (0 through 4 girls) for 4 children with an equal probability of being born. There's a .0625 probability of no girls, a .25 probability of 1 girl, etc.

= Advanced (for programmers) =

The binompdf( and [[TI-BASIC:Binomcdf|Binomcdf(]] commands are the only ones apart from [[TI-BASIC:Seq_List|seq(]] that can return a list of a given length, and they do it much more quickly. It therefore makes sense, in some situations, to use these commands as substitutes for seq(.

Here's how to do it:

# cumSum(binomcdf(N,0 gives the list {1 2 ... N+1}, and cumSum(not(binompdf(N,0 gives the list {0 1 2 ... N}.
# With seq(, you normally do math inside the list: for example, seq(3I^^2^^,I,0,5
# With these commands, you do the same math outside the list: 3Ans^^2^^ where Ans is the list {0 1 ... 5}.

An example:

:seq(2^I,I,1,5
can be
:cumSum(binomcdf(4,0
:2^Ans
which in turn can be
:2^cumSum(binomcdf(4,0

In general (where f() is some operation or even several operations):

:seq(f(I),I,1,N
can be
:cumSum(binomcdf(N-1,0
:f(Ans
which can sometimes be
:f(cumSum(binomcdf(N-1,0

If the lower bound on I in the seq( statement is 0 and not 1, you can use binompdf( instead:

:seq(f(I),I,0,N
can be
:cumSum(not(binompdf(N,0
:f(Ans
which can sometimes be
:f(cumSum(not(binompdf(N,0

This will not work if some command inside seq( can take only a number and not a list as an argument. For example, seq(L,,1,,(I),I,1,5 cannot be optimized this way.

= Formulas =

The value of binompdf( is given by the formula

[[TI-BASIC:Math|Math]]
\operatorname{binompdf}(n,p,k) = \binom{n}{k}\,p^k\,(1-p)^{n-k} = \frac{n!}{k!\,(n-k)!}\,p^k\,(1-p)^{n-k}
[[TI-BASIC:/math|/math]]

This formula is fairly intuitive. We want to know the probability that out of n trials, exactly k will be successes, so we take the probability of k successes - <math>p^k</math> - multiplied by the probability of (n-k) failures - <math>(1-p)^{n-k}</math> - multiplied by the number of ways to choose which k trials will be successes - <math>\binom{n}{k}</math>.

= Error Conditions =

* '''[[TI-BASIC:Errors#domain|ERR:DOMAIN]]''' is thrown if the number of trials is at least 1 000 000 (unless the other arguments make the problem trivial)

= Related Commands =

* [[TI-BASIC:Binomcdf|Binomcdf(]]
* [[TI-BASIC:Geometpdf|Geometpdf(]]
* [[TI-BASIC:Geometcdf|Geometcdf(]][[Category:TI-BASIC]]
[[Category:TIBD]]

TI-BASIC:Binompdf - Revision history

Maintenance script: Automated superscript correction

KermMartian at 19:20, 24 February 2016

Maintenance script: Initial automated import