Question

Motorcycle riders wear high visibility clothing so that drivers can spot them easily. 50% of Motorcyclist wear high visibility clothing. A researcher experiments and determines that if drivers see a Motorcyclist, then the probability the Motorcyclist was wearing high visibility clothing was 0.51. That experiment shows that 92% of Motorcyclist were seen by drivers on the road.

A. Find the probability that a driver does not see a Motorcyclist , give that the Motorcyclist was wearing visibility clothing?

B. Find the probability that a driver does not see a Motorcyclist , given that the Motorcyclist was not wearing high visibility clothing

C. If 200 drivers pass a Motorcyclist wearing high visibility clothes, what's the probability that at least one driver doesn't see the Motorcyclist.

Answer 1

Let event A be that the Motorcyclist wear high visibility clothing (Hence event A' be that the Motorcyclist do not wear high visibility clothing)

and event B be drivers see a Motorcyclist (Hence event B' be that the drivers do not see a Motorcyclist)

The probability values given are as follows:

P(A) = 0.50, so P(A')=1-0.50 = 0.50

P(A | B) = 0.51, so P(A'|B) = 1 - 0.51 = 0.49

P(B) = 0.92 therefore, P(B') = 0.08

(a) P(driver does not see a Motorcyclist | Motorcyclist was wearing visibility clothing) = P(B' | A)

Now we know P(A) = P(A|B)*P(B) + P(A|B')*P(B')

P(A|B') = [ P(A) - P(A|B)*P(B) ] / P(B')

Thus, P(A|B') = [0.5 - 0.51*0.92] / 0.08 = 0.385

Thus, P(A intersection B') = P(A|B')*P(B') = 0.385*0.08 = 0.0308

Thus, P(B'|A) = P(A intersection B') / P(A) = 0.0308/0.5 = 0.0616

Thus, P(driver does not see a Motorcyclist | Motorcyclist was wearing visibility clothing) = P(B' | A) = 0.0616

(b) P(driver does not see a Motorcyclist | Motorcyclist was not wearing visibility clothing) = P(B' | A')

Similarly as above,

P(A') = P(A'|B)*P(B) + P(A'|B')*P(B')

P(A'|B') = [ P(A') - P(A'|B)*P(B) ] / P(B')

Thus, P(A'|B') = [0.5 - 0.0.49*0.92] / 0.08 = 0.615

Thus, P(A' intersection B') = P(A'|B')*P(B') = 0.615*0.08 = 0.0492

Thus, P(B'|A') = P(A' intersection B') / P(A') = 0.0492/0.5 = 0.0982

(c) It is assumed that each driver sees or not sees a motorcyclist independent. Also,

P(Driver sees a motorcyclist) = 0.92

P(Driver does not see a motorcyclist) = 0.08

Thus the probability that out of the 200 drivers atleast one driver doesn't see the Motorcyclist is given as

1-P(all the drivers sees the motorcyclist)

1-[ P(Driver 1 see the motor cysclist) * P(Drvier 2 see the motor cysclist) *...* P(Driver 200 see the motor cysclist)]

1-0.92*0.92*...*0.92

1-0.92200

=0.999999

Thus, the probability that at least one driver doesn't see the Motorcyclist is 0.999999