Important help on the subject of polynomials of binomial

[math12345]

Hi, everyone
It is known that binomial distribution can also be solved by polynomials. i add document with a question I can not solve.

Glad to get for help
I have the all chart in Excel but Excel format not be uploaded here
Thanks to all the respondents

we have in the basket has 10 balls
6 red, 4 yellow and the game is played with repetition

1- As soon we has appear 4 time yellow balls in the during of the game the game is finshed, we can played the game max 10 time and it over. What is the probability that the game will end between 4 and 10?

2-In which game is the probability is highest that we will get the first value,The second value, the third value, the fourth value yellow ball?

3- Repeating the same action without repetition What was the effect on the results?
we have in the basket has 10 balls
6 red, 4 yellow and the game is played with repetition

1- As soon we has appear 4 time yellow balls in the during of the game the game is finshed, we can played the game max 10 time and it over. What is the probability that the game will end between 4 and 10?

2-In which game is the probability is highest that we will get the first value,The second value, the third value, the fourth value yellow ball?

3- Repeating the same action without repetition What was the effect on the results?
we have in the basket has 10 balls
6 red, 4 yellow and the game is played with repetition

1- As soon we has appear 4 time yellow balls in the during of the game the game is finshed, we can played the game max 10 time and it over. What is the probability that the game will end between 4 and 10?

2-In which game is the probability is highest that we will get the first value,The second value, the third value, the fourth value yellow ball?

3- Repeating the same action without repetition What was the effect on the results?

MY Request for help with the following next things



I understand that we are dealing with recursion - retreat series

How to answer questions 1 2 3 from a game table I performed -

Why in stage 1-4 the probability was equal to 1 and in stage 5 the probability -
decreases by the During each round

Maybe it is necessary to make some connection between steps in the- calculations?

The most important questions for me

1. It is known that the same binomial array can be solved by a polynomial and there is an example on the leftI understand need to connect here between a factor and a factor fails to understand what to what and how do you know if it is a probability of a red or yellow ball? And how for God's sake cause the 2 arrays to have the same result in a different way

Thank you very much to all the respondents

[Guest]

There are 10 balls, four of them yellow, sampled with replacement, so the probability of a yellow ball on any one sample is 4/10= 2/5. There are 6 red balls so the probability of a red ball on any one sample is 6/10= 3/5. The probability of four yellow balls on the first four samples is (2/5)^4. The probability of the first ball being red and the next 4 yellow is (3/5)(2/5)^4. There are 4 permutations of RYYYY, YRYYY, YYRYY, YYYRY (we do not include YYYYR because that would be 4 yellow in the first four samples and there would not be a fifth sample) so the probability of getting 4 yellow in 5 samples is 4(3/5)(2/5)^4.

[math12345]

First thanks for the response

Considering that your description is what I came up with according to the probability formula, and according to polynomials and it does not match

Refers to the following formula

[url]https://www.pinterest.com/pin/314407617711045730/
[/url]

The probability of getting a yellow ball 0 in game 1 0.6 1 0.6
The probability of getting a yellow ball 1 in game 1 0.4 1 0.4
ball 2
The probability of getting a yellow ball 0 in game 2 0.36 1 0.36
The probability of getting a yellow ball in game 1 0.24 2 0.48
The probability of getting a yellow ball 2 in game 2 0.16 1 0.16
ball 3
The probability of getting a yellow ball 0 in game 3 0.216 1 0.216
The probability of getting a yellow ball 1 in game 3 0.144 3 0.432
The probability of getting a yellow ball 2 in game 3 0.096 3 0.288
The probability of getting a yellow ball 3 in game 3 0.064 1 0.064
ball 4
The probability of getting a yellow ball 0 in game 4 0.1296 1 0.1296
The probability of getting a yellow ball 1 in game 4 0.0864 4 0.3456
The probability of getting a yellow ball 2 in game 4 0.0576 6 0.3456
The probability of getting a yellow ball 3 in game 4 0.0384 4 0.1536
The probability of getting a yellow ball 4 in game 4 0.0256 1 0.0256
ball 5
The probability of getting a yellow ball 0 in game 5 0.07776 1 0.07776
The probability of getting a yellow ball 1 in game 5 0.05184 5 0.2592
The probability of getting a yellow ball 2 in game 5 0.03456 10 0.3456
The probability of getting a yellow ball 3 in game 5 0.02304 10 0.2304
The probability of getting a yellow ball 4 in game 5 0.01536 5 0.0768
The probability of getting a yellow ball 4 in game 5 0.01536 5
ball 6
The probability of getting a yellow ball 0 in game 6 0.046656 1 0.046656
The probability of getting a yellow ball 1 in game 6 0.031104 6 0.186624
The probability of getting a yellow ball 2 in game 6 0.020736 15 0.31104
The probability of getting a yellow ball 3 in game 6 0.013824 20 0.27648
The probability of getting a yellow ball 4 in game 6 0.009216 15 0.13824
The probability of getting a yellow ball 4 in game 6 0.009216 15
The probability of getting a yellow ball 4 in game 6 0.009216 15
ball 7
The probability of getting a yellow ball 0 in game 7 0.0279936 1 0.0279936
The probability of getting a yellow ball 1 in game 7 0.0186624 7 0.1306368
The probability of getting a yellow ball 2 in game 7 0.0124416 21 0.2612736
The probability of getting a yellow ball 3 in game 7 0.0082944 35 0.290304
The probability of getting a yellow ball 4 in game 7 0.0055296 35 0.193536
The probability of getting a yellow ball 4 in game 7 0.0055296 35
The probability of getting a yellow ball 4 in game 7 0.0055296 35
The probability of getting a yellow ball 4 in game 7 0.0055296 35
ball 8
The probability of getting a yellow ball 0 in game 8 0.01679616 1 0.01679616
The probability of getting a yellow ball 1 in game 8 0.01119744 8 0.08957952
The probability of getting a yellow ball 2 in game 8 0.00746496 28 0.20901888
The probability of getting a yellow ball 3 in game 8 0.00497664 56 0.27869184
The probability of getting a yellow ball 4 in game 8 0.00331776 70 0.2322432
The probability of getting a yellow ball 4 in game 8 0.00331776 70
The probability of getting a yellow ball 4 in game 8 0.00331776 70
The probability of getting a yellow ball 4 in game 8 0.00331776 70
The probability of getting a yellow ball 4 in game 8 0.00331776 70
ball 9
The probability of getting a yellow ball 0 in game 9 0.010077696 1 0.010077696
The probability of getting a yellow ball 1 in game 9 0.006718464 9 0.060466176
The probability of getting a yellow ball 2 in game 9 0.004478976 36 0.161243136
The probability of getting a yellow ball 3 in game 9 0.002985984 84 0.250822656
The probability of getting a yellow ball 4 in game 9 0.001990656 126 0.250822656
The probability of getting a yellow ball 4 in game 9 0.001990656 126
The probability of getting a yellow ball 4 in game 9 0.001990656 126
The probability of getting a yellow ball 4 in game 9 0.001990656 126
The probability of getting a yellow ball 4 in game 9 0.001990656 126
The probability of getting a yellow ball 4 in game 9 0.001990656 126
The probability of getting a yellow ball 4 in game 9 0.001990656 126
ball 10
The probability of getting a yellow ball 0 in game 10 0.006046618 1 0.006046618
The probability of getting a yellow ball 1 in game 10 0.004031078 10 0.040310784
The probability of getting a yellow ball 2 in game 10 0.002687386 45 0.120932352
The probability of getting a yellow ball 3 in game 10 0.00179159 120 0.214990848
The probability of getting a yellow ball 4 in game 10 0.001194394 210 0.250822656
The probability of getting a yellow ball 4 in game 10 0.001194394 210
The probability of getting a yellow ball 4 in game 10 0.001194394 210
The probability of getting a yellow ball 4 in game 10 0.001194394 210
The probability of getting a yellow ball 4 in game 10 0.001194394 210
The probability of getting a yellow ball 4 in game 10 0.001194394 210
The probability of getting a yellow ball 4 in game 10 0.001194394 210

[math12345]

Doubling and result combination polynomial

1.24 1 1.24 X^1+1x*a+a^1= (x+a)^n



1.24 1 1.24 X^1+1x*a+a^1= (x+a)^n



1 1 1 X^2+2x*a+a^2+ (x+a)^n



2 2 1 X^2+2x*a+a^2= (x+a)^n



1 1 1 X^2+2x*a+a^2= (x+a)^n



1 1 1 X^3+3x*a+a^3= (x+a)^n



3 3 1 X^3+3x*a+a^3= (x+a)^n



3 3 1 X^3+3x*a+a^3= (x+a)^n



1 1 1 X^3+3x*a+a^3= (x+a)^n