Probability: Draw without replacement

$P_{success}= \frac{ ^nC_r \times ^pP_r \times ^{(s-p)}P_{(n-r)}}{^sP_n}$

$= \dfrac{ ^pC_r \times ^{(s - p)}C_{(n - r)} }{^sC_n}$

$n$ =number of draws

$p$ =maximum number of successful draws

$s$ =maximum number of draws

$r$ = number of successful draws obtained

(still incomplete) It's similar to the probability of Bernoulli trial

Bernoulli trial: either pass or fail, but with replacement

$P_{success} = \text{ }^nC_r \times p^r \times (1-p)^{n-r}$

$^nC_r \text{ represents the total number of configurations of success and failure from the obtained ones}$

$^pP_r \text{ represents the total number of successful choices}$

$^{(s-p)}P_{(n-r)} \text{ represents the total number of unsuccessful choices}$

$^sP_n \text{ represents the total number of combinations that can be obtained for } n \text{ draws}$

This is also called a hypergeometric distribution

#Combinatorics

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Probability: Draw without replacement

Psuccess=nCr×pPr×(s−p)P(n−r)sPn P_{success}= \frac{ ^nC_r \times ^pP_r \times ^{(s-p)}P_{(n-r)}}{^sP_n}Psuccess​=sPn​nCr​×pPr​×(s−p)P(n−r)​​

Comments

$P_{success}= \frac{ ^nC_r \times ^pP_r \times ^{(s-p)}P_{(n-r)}}{^sP_n}$