Banner Ad 1

Collapse

Announcement

Collapse
No announcement yet.

are really the events independent?

Collapse
This topic is closed.
X
X
 
  • Filter
  • Time
  • Show
Clear All
new posts

  • are really the events independent?

    Hello all!

    I am currently preparing for exam p and I would really appreciate it if someone could answer the following question about binomial distribution:

    There are questions where I am given the size of the group, say 100, and I am given that 31% , for example approve of the internet . Then I am asked what the probability of having 5 out of a random sample of 10 people to approve of the Internet is. Of course the answer is (10 choose 5)(0.31)^{5}(0.69)^{5}. However, my question is - are really the events here independent. If I pick one person and he approves of the internet, doesn't this mean that I am changing the probability that the next one either approves or not of the internet?

    I really hope that you can help me!

  • #2
    I hope someone answers you soon. I believe it depends on how the question is posed to you and whether or not sampling with or without replacement is specified in the question but I see where the confusion may lie and I'm wondering the same thing.

    Comment


    • #3
      Originally posted by thecool View Post
      Hello all!

      I am currently preparing for exam p and I would really appreciate it if someone could answer the following question about binomial distribution:

      There are questions where I am given the size of the group, say 100, and I am given that 31% , for example approve of the internet . Then I am asked what the probability of having 5 out of a random sample of 10 people to approve of the Internet is. Of course the answer is (10 choose 5)(0.31)^{5}(0.69)^{5}. However, my question is - are really the events here independent. If I pick one person and he approves of the internet, doesn't this mean that I am changing the probability that the next one either approves or not of the internet?

      I really hope that you can help me!
      Zeigy is correct. Say you have 100 marbles in a bag and 10% of them are red. You put your hand and take out a marble, whatever it is, red or not and then you throw it back in. Now you draw again, is there any more or less probability that you will pick out a red marble (or a non-red one)?

      Comment


      • #4
        I reviewed binomial distribution this morning.

        Okay. Your question is basically when would one use the binomial distribution and when does one use the hypergeometric distribution.

        Yes you are changing the probability that the next person you choose either approves or not. However, if the population is large enough relative to the sample size you are taking then the probability changes so slightly that it doesn't adversely affect the result. For example if you had to randomly choose 10 persons from a population size of 1 million. The probability that the first person you choose approves of the internet is .31000000. The probability that the second person you choose approves of the internet given that the first does is 0.30999931. The third person, 0.30999862. As you can see, the probability changes very little and by the time you round off to two decimal places all the probabilities are 0.31.

        Now your sample size was 10 persons from a population of 100. The probability that the first person you randomly choose approves of the internet is 0.310000000. The second person given the first does is 0.303030303. The third given another success is 0.295918367. Let's say on the fourth you randomly chose someone who did not approve, you've reduced the pool yet further (you've reduced the pool of 100 already by three from the first three trials from 100 to 97), the fifth and sixth trials also had persons who did not approve, now the eight trial you randomly chose someone who did approve, that probability is 0.308510638.

        Hopefully you get the idea how the probabilities rise and fall as you randomly choose successes and failures from the pool of 100. As a good rule of thumb if the sample size is 10% or less of the population size, as it is in this question, then the binomial distribution would give a fair result. Otherwise you would need to use the hypergeometric distribution to account for the changing probabilities as you sample without replacement.

        Solving your binomial question using the hypergeometric distribution should give us a similar (more accurate) result.

        (31C5 x 69C5) / 100C10 = 0.1103127

        Your binomial solution is 0.112837762

        In both cases the solution is approximately 11%.
        Last edited by Zeigy; July 16 2011, 01:16 PM. Reason: Spelling error.

        Comment


        • #5
          Thanks. I understand what you are saying. I also thought that binomial distribution is used when the probability changes slightly, but I was just not sure whether I was right.

          Comment

          Working...
          X