ToughSTEM
Sign Up
Log In
ToughSTEM
A question answer community on a mission
to share Solutions for all STEM major Problems.
Cant find a problem on ToughSTEM?
0

Police often set up sobriety checkpoints-roadblocks where drivers are asked a few brief questions to allow the officer to judge whether or not the person may have been drinking. If the officer does not suspect a problem, drivers are released to go on their way. Otherwise drivers are detained for a Breathalyzer test that will determine whether or not they will be arrested. Trained officers can make the right decision 80% of the time based on the initial brief stop. Suppose the police operate such a checkpoint on a Saturday night after 9pm, a time when national traffic safety experts suspect that about 12% drivers have been drinking. (Draw a tree diagram if needed)

a) You are stopped at the checkpoint and, of course, have not been drinking. What's the probability that you are detained for further testing?

b) What's the probability that any given driver will be detained?

c) What's the probability that a driver who is detained has actually been drinking?

d) What's the probability that driver who was released had actually been drinking?

Edit
Edited Mon, 26 Sep '16
Added Mon, 26 Sep '16
Comment
Add a Comment
Solutions
0

D = Detained

K = Drinking

Right 80% of the time → P(D|K) = P(D'|K') = 0.8

(It could imply that P(D∩K) + P(D'∩K') = 0.8, but this would lead to insufficient data to solve the problem)

a) P(D|K') = P(D∩K') / P(K') = 0.2 * 0.88 / 0.88 = 0.2

Note, P(D|K') = 1 - P(D'|K') in this case particularly.

b) P(D) = ∑P(D∩$$K_i$$) = ∑P($$K_i$$)*P(D|$$K_i$$) = 0.12 * 0.8 + 0.88 * 0.2 = 0.272

c) P(K|D) = P(K∩D) / P(D) = P(D|K)P(K)/P(D) = 0.8*0.12/0.272 = 0.3529

d) P(K|D') = P(K∩D') / P(D') = P(K∩D') / (∑P($$K_i$$)P(D'|$$K_i$$) = 0.12*0.2 / (0.12*0.2 + 0.88*0.8) = 0.024 / 0.728 = 0.03296

Edit
Edited Mon, 26 Sep '16
Added Mon, 26 Sep '16
Comment
Add a Comment
Add Your Solution!
Close

Click here to
Choose An Image
or
Get image from URL
GO
Close
Back
Add Image
Close
What URL would you like to link?
GO
α
β
γ
δ
ϵ
ε
η
ϑ
λ
μ
π
ρ
σ
τ
φ
ψ
ω
Γ
Δ
Θ
Λ
Π
Σ
Φ
Ω
Copied to Clipboard

Add Your Solution
Sign Up
to interact with the community. (That's part of how we ensure only good content gets on ToughSTEM)
OR
OR
ToughSTEM is completely free, and its staying that way. Students pay way too much already.
Almost done!
Please check your email to finish creating your account!
Welcome to the Club!
Choose a new Display Name
Only letters, numbers, spaces, dashes, and underscores, are allowed. Can not be blank.
Great! You're all set, .
A question answer community on a mission
to share Solutions for all STEM major Problems.
Why
The Purpose
How
The Community
Give Feedback
Tell us suggestions, ideas, and any bugs you find. Help make ToughSTEM even better.