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February 25th, 2019, 09:12 PM  #1 
Member Joined: Aug 2018 From: România Posts: 90 Thanks: 6  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. 
February 26th, 2019, 11:22 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,590 Thanks: 1434 
1) $p = \dfrac{25}{40}=\dfrac{5}{8}$ $1p = \dfrac{3}{8}$ $P[\text{3 consecutive white balls}] = \sum \limits_{w=3}^5 P[\text{3 consecutive white balls}w \text{ white balls chosen}] $ $P[3c3] = 3\cdot p^3 (1p)^2$ $P[3c4] = 4 \cdot 2 p^4 (1p)$ $P[3c5] = p^5$ $P[\text{3 consecutive white balls}] = 3p^3 (1p)^2+4p^4 (1p)+p^5$ I leave it to you to crunch the numbers. 
March 4th, 2019, 08:52 PM  #3  
Member Joined: Aug 2018 From: România Posts: 90 Thanks: 6  Quote:
Please give details.  It does not have to be $\displaystyle P[3c4] = 4 \cdot p^4 (1p)$? It does not have to be $\displaystyle P[3c5] = 5p^5$? Thank you very much! All the best, Integrator  

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