Probability question on questionnaire

[mjordan2nd]

Homework Statement

When sent a questionnaire, 50% of the recipients respond immediately. Of those who do not respond, 40% respond when sent a follow-up letter. If the questionnaire is sent to 4 persons and a follow-up letter is sent to any of the 4 who do not respond immediately, what is the probability that at least 3 never respond?

Homework Equations

$$P\left[ \geq \mbox{ 3 never respond} \right] = P \left[ \mbox{none respond} \right] + P \left[ \mbox{1 responds, 3 don't} \right]$$

The Attempt at a Solution

The probability of any individual not responding is 0.3. So

$$P \left[ \mbox{none respond} \right] = (0.3)^4$$.

On the other hand

$$P \left[ \mbox{1 responds, 3 don't} \right] = P \left[ \mbox{1 response on 1st round, 0 on second} \right] + P \left[ \mbox{ 0 responses on 1st round, 1 response on 2nd} \right ]$$

$$P \left[ \mbox{1 responds, 3 don't} \right] = (.5)^4(.6)^3 + (.5)^4(.6)^3(.4) = (.3)^3(.7).$$

Therefore, based on what I calculate my answer should be

$$P\left[ \geq \mbox{ 3 never respond} \right] = (.3)^4 + (.3)^3(.7).$$

However, the closest answer choice I have available is

$$P\left[ \geq \mbox{ 3 never respond} \right] = (.3)^4 + 4(.3)^3(.7),$$

where the second term in the answer has a factor of 4 that I did not get. My guess is I'm getting confused somewhere and only getting the probability for one person and not four, but I'm having a hard time seeing why this is. Why is their a factor of 4 on the second term? And if that's supposed to be there, why is there not another factor of 4 on the first term? Thanks.

 

[da_nang]

What's the probability of the first person responding and the others don't? What's the probability of the second person responding and the others don't?

You should be able to notice something from those answers.

 

[Orodruin]

Apart from what da_nang just said, the probability of any particular person responding is 0.3 as you have seen. You do not need to go through the middle steps of splitting the cases into who responds when.

 

[HallsofIvy]

50% of the people respond immediately. Of the 50% who don't, 40%, or (.5)(.4)= .2 so 20% of the original number respond on a second mailing. That is a total of 70% of the original number. Now it becomes a "binomial" distribution with p= 0.7 and q= 1- p= 0.3.

 

[Ray Vickson]

50% of the people respond immediately. Of the 50% who don't, 40%, or (.5)(.4)= .2 so 20% of the original number respond on a second mailing. That is a total of 70% of the original number. Now it becomes a "binomial" distribution with p= 0.7 and q= 1- p= 0.3.

Or, you can look at the non-responders, whose first- and second-stage non-response probabilities are 0.5 and 0.6, giving an overall p = 0.30 of never responding.