Explain and justify the problems below with steps: DUI : An individual suspected
ID: 3355092 • Letter: E
Question
Explain and justify the problems below with steps:
DUI: An individual suspected of driving under the influence of alcohol may be given a “rapid test” which has the probability of .75 of being correct. Assume that 60% of the drivers stopped for being suspected of driving under the influence of alcohol actually are. The other 40% are just sleepy or texting on their cell phones.
a) What is the probability that the next 4 drivers stopped for this suspicion will all be under the influence of alcohol?
b) If Hivebin Drunkbefor tests positive, what is the chance that he actually has been driving under the influence?
If this test gives a positive result, the individual is subjected to a further test that detects a DUI with a 90% chance of being accurate. It is 100% accurate for those having an alcohol content below the legal limit. Assume the two tests are independent.
c) What proportion of subjects will pass the second test?
d) What is the probability that a suspect with alcohol content above the legal limit will pass the second test?
e) What proportion of subjects are not given the second test?
f) What is the probability that a suspect had an alcohol content over the legal limit if he is not given the second test?
Explanation / Answer
Answer to the question)
given that the test is correct only 75% of the times
People actually under the influence of alcohol = 60%
People not under the influence of alcohol = 40%
Thus people who test positive and are under influence of alcohol = 75%*60% = 45%
people who test negative and are under influence of alcohol = 25%*60% = 15%
People who test negative and are not under the influenc eof alcohol = 75% * 40% = 30%
People who test Positive and are not under the influenc eof alcohol = 25% *40% = 10%
Total Positive tests = 45%+10% = 55%
Total negative tests = 30%+15% = 45%
.
Answer to part a)
P(next four dirvers are actually under the influence of alcohol) = P(first driver) * P(second driver) * P(thrid driver) * P(fourth driver)
[we multiply each of the probability values because these are independent events, whether second driver is under the influence or not , does not depend on any other driver's state]
P(next four drivers are actually under the influence of alcohol) = 0.6*0.6*0.6*0.6 = 0.1296
.
Answer to part b)
P(positive result) = 55% = 0.55
P(actually under the infuence and test positive) = 0.45 ( as calculated above)
P( under the infuence of alcohol | test positive) = P(under the influence and test positive) / P(test positive)
[This is the conditional formula of probability]
P( under the infuence of alcohol | test positive) = 0.45 /0.55
P( under the infuence of alcohol | test positive) = 0.8182
.
Answer to part c)
All those who actually had alcohol the next test is 100% correct
thus those who had alcohol and tested positive in first test = 45%
for them the test is working completely , thus 100% * 45% = 45% pass the test
Thoe who didnot have alcohol but passed the test = 10%
for these 10% of the people, the test is 90% accurate
This implies still 10% of these people wll be showing the positive result ,t hough they are not under the influence of alcohol , thus percent of positives = 10% * 10% = 1%
Hence total percent of people who test positive in second test = 45% + 1% = 46%
Total percent of peopl who test negative in second tets = 9%*10% = 9%
.
Answer to part d)
The probability that a person with alcohol content above legal limit will pass the test is given as 100%
.
Answer to part e)
The people who tested negative in the first test were not given the second test
Thus 25% of 60% people who actually were under the influence &
75% of 40% people who were not under the influence of alcohol , actually tested negative in the first test
Hence , total = 25% * 60% + 75% * 40% = 15% +30% = 45%
Thus 45% of total people tested negative in first test and thus were not given the second test
.
Answer to part f)
Probability of people who were not given the second test = 45% or 0.45
out of them ,the probability that a suspect had alcohol but tested negative under first test = 25% of 60%
Thus 25% of 60% = 15% of the people were not given the second test & they had alcohol above the legal limit
Thus P(had alcohol | not given second test) = P(had alcohol & not given second test) / P(not given second test)
[this is the conditional formula of probability used]
P(had alcohol | not given second test) = 0.15 /0.45
P(had alcohol | not given second test) = 0.3333
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