## Wednesday, August 31, 2016

### HP 12C: Combination/Binomial Distribution/Negative Binomial Distribution

HP 12C:  Combination/Binomial Distribution/Negative Binomial Distribution

Introduction and Formulas

Combination: Find the number of groups out of a possible set of objects.  The order of objects obtained does not matter.
Store n in R1, x in R0, and p in R2.  Press [ f ] [ R↓]  (CLEAR PRGM), [ R/S ]
Formula:  COMB(n, x)  = n!/(x! * (n-x)!)

Binomial Distribution:  Find number of successes (x) in a fixed number of trials (n).
Store n in R1, x in R0, and p in R2.  Press [ g ] [ R↓ ] (GTO) 26, [R/S]
Formula:  COMB(n, x) * p^x * (1 – p)^(n – x)

Negative Binomial Distribution:  Find the number of trials (n) needed to obtain a fixed amount of successes (x).
Store x in R1, n in R0, and p in R2.  Press [ g ] [ R↓ ] (GTO) 43, [R/S]
Formula:  COMB(x – 1, n – 1) * p^(n -1) * (1 – p)^((x - 1) - (n – 1))

In the distribution calculations, p is the probability where 0 ≤ p ≤ 1.
Note: R3 is used as a flag, which will allowed for branching.

 STEP CODE KEY Combination 01 0 0 02 44, 3 STO 3 03 45, 1 RCL 1 04 43, 3 N! 05 45, 0 RCL 0 06 43, 3 N! 07 10 ÷ 08 45, 1 RCL 1 09 45, 0 RCL 0 10 30 - 11 43, 3 N! 12 10 ÷ Flag Testing 13 45, 3 RCL 3 14 1 1 15 30 - 16 43, 35 X=0 17 43, 33, 29 GTO 29 18 45, 3 RCL 3 19 2 2 20 30 - 21 43, 35 X=0 22 43, 33, 49 GTO 49 23 33 R↓ 24 33 R↓ 25 43, 33, 00 GTO 00 Binomial Distribution 26 1 1 27 44, 3 STO 3 28 43, 33, 03 GTO 03 29 33 R↓ 30 45, 2 RCL 2 31 45, 0 RCL 0 32 21 Y^X 33 20 * 34 1 1 35 45, 2 RCL 2 36 30 - 37 45, 1 RCL 1 38 45, 0 RCL 0 39 30 - 40 21 Y^X 41 20 * 42 43, 33, 00 GTO 00 Negative Binomial Distribution 43 1 1 44 44, 30, 1 STO- 1 45 44, 30, 0 STO- 0 46 2 2 47 44, 3 STO 3 48 43, 33, 03 GTO 03 49 33 R↓ 50 33 R↓ 51 45, 2 RCL 2 52 45, 0 RCL 0 53 21 Y^X 54 20 * 55 1 1 56 45, 2 RCL 2 57 30 - 58 45, 1 RCL 1 59 45, 0 RCL 0 60 30 - 61 21 Y^X 62 20 * 63 43, 33, 00 GTO 00

Examples:

Find the number of combinations of groups of 2 out of possible 12 objects.
12 [STO] 1, 2 [STO] 0, [ f ] [ R↓ ] (CLEAR PRGM)
Result:  66

Binomial Distribution:  Toss a coin 25 times. (trails) What is the probability of tossing 10 heads? (successes)  Assume a fair coin.  The variables n = 25, x = 10, p = 0.5
25 [ STO ] 1, 10 [ STO ] 0, 0.5 [ STO ] 2, [ g ] [ R↓ ] (GTO) 26 [ R/S ]
Result:  0.10   (0.0974166393)

Negative Binomial Distribution:  Assume a fair coin. What is the probability that the 15th tossed of heads comes on the 25th toss of the coin?  x = 15, n = 25, p = 0.5
25 [STO] 1, 15 [STO] 0, 0.5 [ STO ] 2, [ g ] [ R↓] (GTO) 43 [ R/S ]
Result:  0.12  (0.1168999672)

This blog is property of Edward Shore, 2016.

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