# Thread: Probability Question ( quite challenging)

1. ## Probability Question ( quite challenging)

A husband and wife couple is taking driving tests. If the probability that either the husband or wife passes the test , each time time the test is taken, is 0.8.

i)Find the probability that the husband or the wife passess the test after taking the test exactly twice.

ii)find the probability that both the husband and the wife pass the test after taking test more than two times.

Your help will be pretty much appreciated.

2. ## Probability

Hello ose90
Originally Posted by ose90
A husband and wife couple is taking driving tests. If the probability that either the husband or wife passes the test , each time time the test is taken, is 0.8.

i)Find the probability that the husband or the wife passess the test after taking the test exactly twice.

ii)find the probability that both the husband and the wife pass the test after taking test more than two times.

Your help will be pretty much appreciated.
I assume that the question means that each time the husband takes the test, then the probability that he passes is 0.8; and the same is true of the wife. So the probability that either one fails on a particular test is 0.2. So:

(i) The probability that neither passes on their first test is 0.2 x 0.2 = 0.04. Then, on the second test for each person the probability that one passes and the other fails is 2 x 0.2 x 0.8 = 0.32. So the probability that exactly one passes at the second attempt is 0.04 x 0.32 = 0.0128.

(ii) The probability that they both pass after more than two tests is the same as the probability that they both fail on the first two tests = 0.2 x 0.2 x 0.2 x 0.2= 0.0016.

Hello ose90I assume that the question means that each time the husband takes the test, then the probability that he passes is 0.8; and the same is true of the wife. So the probability that either one fails on a particular test is 0.2. So:

(i) The probability that neither passes on their first test is 0.2 x 0.2 = 0.04. Then, on the second test for each person the probability that one passes and the other fails is 2 x 0.2 x 0.8 = 0.32. So the probability that exactly one passes at the second attempt is 0.04 x 0.32 = 0.0128.

(ii) The probability that they both pass after more than two tests is the same as the probability that they both fail on the first two tests = 0.2 x 0.2 x 0.2 x 0.2= 0.0016.

A big thank for your explanations. What you've written seems to be correct and logical but the answers shown in the answer key are 0.2944 and 0.01067 respectively.

For the first question, I was wondering why couldn't both of them pass the test together in their second attempt. Is it because of the word 'or'? Is it true that my statement would only be plausible if the question states husband and wife pass the exams at second attempt? (In this case,only the event that both pass the test together in their 2nd attempt while both fail in their first attempt are taken into account, right?)

Again, thank you for helping me!

4. ## Probability

Hello again ose90
Originally Posted by ose90
A big thank for your explanations. What you've written seems to be correct and logical but the answers shown in the answer key are 0.2944 and 0.01067 respectively.

For the first question, I was wondering why couldn't both of them pass the test together in their second attempt. Is it because of the word 'or'? Is it true that my statement would only be plausible if the question states husband and wife pass the exams at second attempt? (In this case,only the event that both pass the test together in their 2nd attempt while both fail in their first attempt are taken into account, right?)

Again, thank you for helping me!
The wording of questions of this type is absolutely critical, and you'll notice that I started my solution by giving my interpretation of its meaning. You'll also see that I was careful in part (i) to interpret the word 'or' as an 'exclusive or'; in other words, one or other but not both.

If we allow the 'or' in part (i) to be an 'inclusive or' - in other words one or both pass at the second attempt, then the probability that this occurs is 1 - p(both fail) = 1 - 0.04 = 0.96. In this case, the probability that they both fail at the first attempt, and then one or both pass at the second attempt is 0.04 x 0.96 = 0.0384. Which is, obviously, still wildly different from the answer you have been given!

Are you sure that you have written down the question exactly as it appears? I can't really see what else it could mean, based on what you wrote in your original post. But if the given answers are correct, then we must be misinterpreting the question in some way.

5. Originally Posted by ose90
A big thank for your explanations. What you've written seems to be correct and logical but the answers shown in the answer key are 0.2944 and 0.01067 respectively.

For the first question, I was wondering why couldn't both of them pass the test together in their second attempt. Is it because of the word 'or'? Is it true that my statement would only be plausible if the question states husband and wife pass the exams at second attempt? (In this case,only the event that both pass the test together in their 2nd attempt while both fail in their first attempt are taken into account, right?)

Again, thank you for helping me!
Let H be the test number on which the husband passes, and let W be the test number on which the wife passes. Then

$P(H = 2 \text { or } W = 2) = P(H = 2) + P(W = 2) - P(H = 2 \text { and } W = 2)$
$= 2 (.2)(.8) - [(.2)(.8)]^2 = 0.2944$

As for the second part, I have no explanation for 0.01067-- it seems to me Grandad's answer of (0.2)^4 is correct.

6. ## Probability

Hello everyone
Originally Posted by awkward
Let H be the test number on which the husband passes, and let W be the test number on which the wife passes. Then

$P(H = 2 \text { or } W = 2) = P(H = 2) + P(W = 2) - P(H = 2 \text { and } W = 2)$
$= 2 (.2)(.8) - [(.2)(.8)]^2 = 0.2944$

As for the second part, I have no explanation for 0.01067-- it seems to me Grandad's answer of (0.2)^4 is correct.
My reasoning for (i) was based on the assumption that neither passes on the first test - which is not necessarily so. My initial reading of the question assumed that each person took the test at least twice. Apologies!

I remain as puzzled as awkward as to where the answer to (ii) comes from.