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$\text{*The below problem was asked in geometric distribution section}$

In a population there are $50\%$ Male and $50\%$ Female

What is the probability to find $2$ Females in a row out of $10$ people (i)with Replacement and (ii)without replacement.

Probable solution, I think is as below.

i) $(0.5)^2 * (0.5)^{10}$

ii) $(0.5)^2 * (0.5)^8$

However, I get confuse when at other times I see the probability calculated as

i) $\frac{5}{10} * \frac{5}{10}$

ii) $\frac{5}{10} * \frac{4}{9}$

KittyL
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Shoaibkhanz
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1 Answers1

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You picked $10$ people from a population with half male and half female. You are now picking $2$ people from them.

With replacement, every time there is $50%$ probability to get a female. So the probability is $0.5\cdot 0.5$.

Without replacement, it is like tossing $10$ coins and find the probability of existence of $2$ consecutive heads. You can refer to this post: Probability of tossing a fair coin with at least $k$ consecutive heads.

Community
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KittyL
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  • I can't understand why the assumption that there are 5 men and 5 women in that set of 10 persons. The way I understand the question as asked is that first we randomly choose 10 persons out of that population and then...etc. – Timbuc Apr 25 '15 at 09:58
  • @Timbuc: It is bad-worded. But I think the "with" or "without replacement" phrases meant to say that. – KittyL Apr 25 '15 at 10:06
  • Why would the with/without thing imply that from the 10 persons half are men and half are women? I don't follow...Thanks. – Timbuc Apr 25 '15 at 10:07
  • @Timbuc: Oh I meant that the "with" or "without" implies you have 10 people and you are picking $2$ from them. As for the $5$ and $5$, it comes from the assumption of $50%$ and $50%$. Or strictly speaking I shouldn't say $5$ and $5$, but $50%$ probability instead. Thanks for point that out. – KittyL Apr 25 '15 at 10:11
  • Ok, then we're getting closer perhaps to what I meant: some probability must kick in, or at least some basic assumption, that we have exactly 5 men and 5 women. I know that in this case the problem is seriously more difficult to solve and this seems to be a rather basic thing, yet why wasn't the problem given as "there is a group of 10 persons, 5 of whom are wome and the other 5 men. Then..." etc. ? – Timbuc Apr 25 '15 at 10:13
  • @Timbuc: I see what you mean now. But I cannot remember the theorem leading to this. Could you give a reference? – KittyL Apr 25 '15 at 10:19
  • I'm not sure what theorem are you referring to. In this case, I'd go by first evaluating the probability that in that set of 10 persons there are at least 2 women (in case of not replacement), and at least one if there's replacement, and then make the probability calculation in each case. But this looks pretty messy, and perhaps some Bayes Theorem (extended form) is necessary here. Anyway, I think the problem itself is a little unclear on this... – Timbuc Apr 25 '15 at 10:23
  • @Shoaibkhanz: Sorry I agree with Timbuc. The problem could be much harder than this. – KittyL Apr 25 '15 at 10:34