1

can you help me with this problem of statistics?

" Consider the experiment of extraction of $3$ balls from an urn containing 16 red balls, $19$ yellow and $5$ blue.

1) Compute the probability of extracting zero red balls, in the event that extractions are carried out without replacement.

Consider, now, the experiment of $50$ draws with replacement.

2) Calculate the probability of getting at least $31$ balls of blue color.

I tried to solve this problem, and I think answer to question 1). is $24/40\times 23/39\times 22/38 = 253/741$. I don't know if it's right, though.

I also tried to find an answer to question 2. but could not find it.

Thanks to anyone who can help me :)

drhab
  • 151,093
Blackie
  • 13

1 Answers1

0

Your answer of the first question is correct. By the second you are having $50$ experiments that are independent and alike. This because of the replacement. Every experiment can succeed (the ball drawn turns out to be blue, which happens with probability ...?) or fail. That smells 'binomial', doesn't it?

Answer:

$$P\left\{ X\geq31\right\} =\sum_{k=31}^{50}\binom{50}{k}\left(\frac{1}{8}\right)^{k}\left(\frac{7}{8}\right)^{50-k}$$

Note that this equals $1-P\left\{ X\leq30\right\} $ and on calculators normally:

$$P\left\{ X\leq30\right\} =\text{binomialcdf}\left(50,\frac{1}{8},30\right)$$

(or something that looks like it)

drhab
  • 151,093
  • The ball can be blue with probability of 5/40 = 1/8. That is if the experiment succeed. If it doesn't then probability is 35/40 = 7/8. – Blackie Jan 27 '14 at 10:19
  • So you are dealing here with a binomial distribution having parameters $n=50$ and $p=1/8$ and you are asked to calculate $P{X\geq 31}$ here. – drhab Jan 27 '14 at 10:21
  • Then if it is a Binomial, it is: p=1/8 and (1-p)=7/8 and n=50 So, P(X>=31) = (binomial coefficien 50 and 31) * (1/8)^31 * (1/9)^19... is that correct? – Blackie Jan 27 '14 at 10:23
  • I don't manage to get it in a comment. See my answer. What your saying looks like it, but do not forget the addition. – drhab Jan 27 '14 at 10:27
  • So, if I understand correctly, I have to do: P(X>=31) = (binomial coefficient 50 and 31) * (1/8)^31 * (7/8)^19 + (binomial coefficient 50 and 32) * (1/8)^32 * (7/8)^18 + ... + (binomial coefficient 50 and 50) * (1/8)^50 * (7/8)^0 – Blackie Jan 27 '14 at 10:35
  • Yes, you understand correctly. – drhab Jan 27 '14 at 10:37
  • Thank you so very much! You're so very kind and polite :) Thanks for your answer. Wish you a good day! – Blackie Jan 27 '14 at 10:38
  • You are very welcome! Have a good day too! – drhab Jan 27 '14 at 11:06