Scientists were trying out a new lie detection test based on a needle. The needle indicates red if it a lie and green if it’s true. For any person lying, there is a 75% chance of the needle pointing towards red. For a person not lying, there is a 90% chance of the needle pointing towards green. It is known that 10% of people who go for the test are lying. If a person who tests to have lied, If the probability that they actually lied is P, then 11P equals?
Answers
Given : The needle indicates red if it a lie and green if it’s true. For any person lying, there is a 75% chance of the needle pointing towards red. For a person not lying, there is a 90% chance of the needle pointing towards green. It is known that 10% of people who go for the test are lying.
If a person who tests to have lied, If the probability that they actually lied is P,
To Find : then 11P equals
Solution:
Probability of red for liar = 0.75
Probability of green for liar = 0.25 ( 1 - 0.75 = 0.25)
Probability of green for true= 0.9
Probability of red for liar = 0.1 ( 1 - 0.9 = 0.1)
Lying = 0.1 not lying = 0.9
a person who tests to have lied,
if lying (0.1)(0.75) = 0.075
if not lying (0.9)(0.1) = 0.09
probability that they actually lied is = (0.075) / (0.075 + 0.09)
= 75/165
= 5/11
P = 5/11
11P = 5
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Solution :-
- Red = Lie .
- Green = True .
Now, For any person lying, we have ,
- Red Probability = 75% = 75/100 = 0.75
- Green Probability = 25% = 25/100 = 0.25
and, For any person not lying, we have ,
- Red Probability = 10% = 10/100 = 0.1
- Green Probability = 90% = 90/100 = 0.9
now, given that ,
- People who go for the test are lying = 10% = 0.1
- P = probability of a person who tests to have actually lied .
then,
→Probability of a person who actually lied is = Number of favourable outcomes / Total Number of outcomes
→ P = (0.1 * 0.75) / (0.1*0.75 + 0.1*0.9)
→ P = 0.075/(0.075 + 0.09)
→ P = 0.075 / 0.165
→ P = 75 / 165
→ P = 5 / 11
hence,
→ Value of 11P = (5/11) * 11 = 5 (Ans.)
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