Question

A cab drives down a pedestrian and leaves the scene. There are two taxi companies in the city: green taxis and blue taxis.

85% of taxis are green, 15% are blue.

A witness has identified the taxi as blue. It is assumed that witnesses are able to identify the correct taxi color in 80% of the cases and mistake 20% of the cases.

What is the likelihood that the taxi involved was blue? What mistakes are common to do in this type of problem? commentary.

Answer 1

We know that 85% of taxis are green, 15% are blue in the city

A witness has identified the taxi as blue.

It is assumed that witnesses are able to identify the correct taxi color in 80% of the cases and mistake 20% of the cases

Therefore the chances of four possible situations

1. There is a Green taxi and correctly identified as Green; the chance = 85% * 80% = 68%
2. There is a Green taxi and mistaken as Blue; the chance = 85% * 20% = 17%
3. There is a Blue taxi and correctly identified as Blue; the chance = 15% * 80% = 12%
4. There is a Blue taxi and mistaken as Green; the chance = 15% * 20% = 3%

Therefore the likelihood that the taxi involved was blue

= Chances of Blue taxi and correctly identified as Blue / (Chances of Blue taxi and correctly identified as Blue + chance a Green taxi and mistaken as Blue)

= 12 %/( 12% + 17%)

= 12% / 29%

= 41.38%

The common mistakes in this type of problem-

• There is a Green taxi and mistaken as Blue
• There is a Blue taxi and mistaken as Green

As the majority (85%) of taxis are green in the city therefore the chances of green taxi is more.