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Pascal Probability

Row
Number

= n

 

  n  

Possible Combinations 2(n-1)

Total
Possible
Combinations

= 2(n-1)

= Σ Coefficients

(a+b)(n-1)
1 1 1 (a+b)0 = 1
2 1    1 2 (a+b)1 = a + b
3 1    2    1 4 (a+b)2 = a2 + 2ab + b2
4 1    3    3    1 8 (a+b)3 = a3 + 3a2b + 3ab2 + b3
5 1    4    6    4    1 16 (a+b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
6 1    5    10    10    5    1 32 (a+b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
7 1    6    15    20    15    6    1 64 (a+b)6 = a6 + 6a5b + 15a4b3 + 20a3b3 + 15a3b4 + 6ab5 + b6
8 1    7    21    35    35    21    7    1 128 (a+b)7 = a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7
   

Pascal's Triangle

   

Binomial Expansion

"a" and "b" represent the two equiprobable outcomes of a paricular trial or event. Examples are heads or tails on the toss of a coin, or the probability of a male or female birth.

"n" represents the Pascal's table row number. It also represents the number of coefficients in the binomial sequence.

(n-1) represents the order of the binomial sequence or power series. It also represents the number of events in subsets of events or trials in a prolonged series of trials. Thus a fifth order binomial series represents 32 trials each with five events or in the example of tossing coins, throwing 5 coins 32 times.

The numbers shown in green also indicate the order of the sequence associated with the rows in which they appear.

2(n-1) is the total number of possible events associated with any order equation. This in turn is equal to the sum of all the binomial coefficients for the series of trials.

For each row "n", the exponents of the variables "a" and "b" in each term of the binomial expansion indicates the number of possible occurrences of each possible outcome "a" or "b", that is, the possible combination of outcomes.

The coefficients of each term in the binomial expansion indicate the number of possible occurences of the combination. (Also called the frequency of occurence). Thus a seventh order binomial expansion has 8 coefficients and 8 possible different combinations of outcomes.

In the fifth order sequence above, 5a4b represents the combination of 4 "heads" with 1 "tails" which should occur 5 times in a sequence of 32 throws, each with 5 coins.

The probability of occurence of any particular combination of outcomes of a series of trials or events is equal to the coefficient corresponding to that combination divided by 2(n-1), the total of possible outcomes.

Pascal's Triangle is a shorthand way of determining the binomial coefficients. The number (coefficient) in any row is the sum of the two numbers directly above it.

 

See more about Pascal

 

 

 

 

 

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