Question

Tom speaks truth in $30\%$ cases and Dick speaks truth in $25\%$ cases. What is the probability that they would contradict each other?

Answer 1

Hint : In this problem related to probability we need to find the probability of Tom and dick speaking truth as well as lie. This is because we need to find the probability of the event when they contradict each other that when one speaks truth other speaks lie; in case of both of them. Their sum will be the probability that we want. Given is the probability of speaking truth is given and probability of speaking lie is one minus the probability of speaking truth.

Complete step by step solution:
Given that tom speaks truth in $30\%$ cases
That is $P\left( {{T_t}} \right) = 30\%$
$P\left( {{T_t}} \right) = 0.30$
Thus probability that tom speaks lie is $P\left( {{{\bar T}_t}} \right) = 1 - P\left( {{T_t}} \right)$
That is $P\left( {{{\bar T}_t}} \right) = 1 - 0.30 = 0.70$
Now that of Dick will also be found with the same method.
Given that Tom speaks truth in $25\%$ cases.
That is $P\left( {{D_t}} \right) = 25\%$
$P\left( {{D_t}} \right) = 0.25$
Thus probability that tom speaks lie is $P\left( {{{\bar D}_t}} \right) = 1 - P\left( {{D_t}} \right)$
That is $P\left( {{{\bar D}_t}} \right) = 1 - 0.25 = 0.75$
Now we have probabilities of the persons.
Now we have to find the probability that they would contradict each other.
That is the probability of Tom speaking truth and Dick lying and vice versa.
So that is given by,
$P\left( {contradict} \right) = P\left( {{T_t}} \right).P\left( {{{\bar D}_t}} \right) + P\left( {{D_t}} \right).P\left( {{{\bar T}_t}} \right)$
Now put the values from the solution above,
$P\left( {contradict} \right) = 0.30 \times 0.75 + 0.25 \times 0.70$
On multiplying we get,
$P\left( {contradict} \right) = 0.2250 + 0.1750$
On adding we get,
$P\left( {contradict} \right) = 0.4$
Thus this is the probability of the situation mentioned in the problem.
This means if they both speak 100 times; they contradict each other 40 times.
So, the correct answer is “0.4”.

Note : Note that we are given the probabilities of speaking truth only. We found that speaking lies just because we are asked to find the probability of contradicting that their speaking should be exactly opposite.
Also note that probability is always either less than or equal to 1. So we found the probability of lying by subtracting the probability of truth from 1.