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(12C) Combination/Binomial Distribution/Negative Binomial Distribution - Eddie W. Shore - 08-31-2016 07:47 PM
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 00Examples:
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)