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February 25th, 2019, 09:12 PM   #1
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Probabilities

Hello,

In a box there are $\displaystyle 25$ white balls and $\displaystyle 15$ red balls.After each extraction, the ball is returned to box.
1) What is the probability of $\displaystyle 3$ white balls coming out consecutively after $\displaystyle 5$ extractions?
2) What is the probability that $\displaystyle 2$ white balls and $\displaystyle 1$ red ball coming out consecutively after $\displaystyle 10$ extractions?
3) How many extractions should be made so that the probability of point 2) is greater than $\displaystyle \frac{1}{2}$?

All the best,

Integrator

Last edited by Integrator; February 25th, 2019 at 09:15 PM.
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February 26th, 2019, 11:22 AM   #2
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1)

$p = \dfrac{25}{40}=\dfrac{5}{8}$
$1-p = \dfrac{3}{8}$

$P[\text{3 consecutive white balls}] =

\sum \limits_{w=3}^5 P[\text{3 consecutive white balls}|w \text{ white balls chosen}] $

$P[3c|3] = 3\cdot p^3 (1-p)^2$

$P[3c|4] = 4 \cdot 2 p^4 (1-p)$

$P[3c|5] = p^5$

$P[\text{3 consecutive white balls}] = 3p^3 (1-p)^2+4p^4 (1-p)+p^5$

I leave it to you to crunch the numbers.
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March 4th, 2019, 08:52 PM   #3
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Quote:
Originally Posted by romsek View Post
1)

$p = \dfrac{25}{40}=\dfrac{5}{8}$
$1-p = \dfrac{3}{8}$

$P[\text{3 consecutive white balls}] =

\sum \limits_{w=3}^5 P[\text{3 consecutive white balls}|w \text{ white balls chosen}] $

$P[3c|3] = 3\cdot p^3 (1-p)^2$

$P[3c|4] = 4 \cdot 2 p^4 (1-p)$

$P[3c|5] = p^5$

$P[\text{3 consecutive white balls}] = 3p^3 (1-p)^2+4p^4 (1-p)+p^5$

I leave it to you to crunch the numbers.
Hello,

Please give details.
--------------------------------------
It does not have to be $\displaystyle P[3c|4] = 4 \cdot p^4 (1-p)$?
It does not have to be $\displaystyle P[3c|5] = 5p^5$?
Thank you very much!

All the best,

Integrator
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