(12C) Combination/Binomial Distribution/Negative Binomial Distribution
08-31-2016, 07:47 PM
Post: #1
 Eddie W. Shore Senior Member Posts: 824 Joined: Dec 2013
(12C) Combination/Binomial Distribution/Negative Binomial Distribution
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.

Code:
 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)
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