1. ## Probability question

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

Here's my (incorrect) solution:

$P(X\geq3)=P(3)+P(4)$

The probability that a person never replies is $0.4*0.5=0.3$

Therefore, $P(4)=0.3^4$

The probability that exactly 3 people never reply is equal to the probability that one person replies immediately and no people respond in the second round plus the probability that one person replies to the follow-up and no people respond in the first round.

The probability that one person replies immediately and no people respond in the second round is:

$P(Q)=0.5*0.3^3$

And the probability that one person replies to the follow-up and 3 people never reply is:

$P(J)=0.5*0.4*0.3^3$

So

$P(3)=P(Q)+P(J)=1.4*0.5*0.3^3$

$P(X\geq 3)=0.3^4+(1.4*0.5*0.3^3)$

However the given solution is

$P(X\geq 3)=0.3^4+4(1.4*0.5*0.3^3)$

I don't understand where the 4 comes from. Any help?

2. ## Re: Probability question

Hey downthesun01.

How many ways can you arrange the individual events for P(3)?

3. ## Re: Probability question

Oh, so the 4 is from the number of combinations of the individual events

I get it now. I don't know how I didn't consider that. Thanks