Series Question (from a probability question)

In summary, the conversation was about a probability question and the expected number of children a couple will have until they have a daughter. The solution involved defining a random variable and using calculus to find the expected number. The conversation also included a discussion about finding geometric series and their derivatives. One participant was able to find the solution online and it was noted that it is important to understand how the solution is derived.
  • #1
mattmns
1,128
6
[SOLVED] Series Question (from a probability question)

EDIT: Found it, never mind.Here is my question:

How do I find what [tex]\sum_{k=1}^{\infty}k(1/2)^k[/tex] is?

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Here is the original question and my work.
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A couple decides to continue to have children until a daughter is born. What is the expected number of children of this couple?
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So suppose P(having a daughter) = 1/2.

Define the random variable X = number of children until a daughter is born.

Then [itex]f_X(x) = (1-1/2)^{x-1}(1/2)[/itex]

So,
[tex]E(X) = \sum_{x = 1}^{\infty}x f_X(x) = \sum_{k=1}^{\infty}k(1/2)(1/2)^{k-1}[/tex]

or as I wrote it above.

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Now I believe everything I have done up this point is correct, however I can't remember how to do series. I have been looking at my calculus book, but I have yet to find what I need. Any hints would be greatly appreciated, thanks!--------------EDIT: Found it online, seems to be a common geometric series, not sure why I couldn't find it in my book.
 
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  • #2
You know how to find f(x)=sum(x^n), right? f'(x)=sum(n*x^(n-1)). Is that enough of a hint?
 
  • #3
It's good to know how it's derived as well.
 
  • #4
Thanks, indeed it is good to know how to derive it.
 

1. What is a series question in probability?

A series question in probability is a type of question where a sequence of events or trials are given, and the probability of a specific outcome or combination of outcomes is asked. This type of question often involves calculating the probability using the multiplication rule or using the concept of independent or dependent events.

2. How do you solve a series question in probability?

To solve a series question in probability, you need to first identify the total number of possible outcomes and the number of desired outcomes. Then, use the appropriate probability formula, such as the multiplication rule, to calculate the probability of the desired outcome or combination of outcomes.

3. What is the multiplication rule in probability?

The multiplication rule in probability states that the probability of two or more independent events occurring together is equal to the product of their individual probabilities. This rule is often used to solve series questions where the outcomes are dependent on each other.

4. Can a series question involve dependent events?

Yes, a series question can involve dependent events. In these types of questions, the outcomes of one event can affect the outcomes of the subsequent events. It is important to carefully consider the relationship between the events and use the appropriate probability formula to solve the question.

5. Is there a difference between a series question and a combination question?

Yes, there is a difference between a series question and a combination question. A series question involves a sequence of events where the order of the outcomes matters, while a combination question involves selecting a group of objects from a larger set without considering the order of the objects. Series questions often involve the use of the multiplication rule, while combination questions often involve the use of combinations or permutations.

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