The probabilities that A and B will tell the truth are 2 / 3 and 4 / 5 respectively . What is the probability that they agree with each other ?

Let A be the event of A will tell truth. B be the event of B tell truth

  $P\left(A\right)=\frac{2}{3},P\left(Ac\right)=1-P\left(A\right)=\frac{1}{3}$

$P\left(B\right)=\frac{4}{5},P\left(Bc\right)=1-P\left(B\right)=\frac{1}{5}$ 

When both agree then they say true or they say false together, that is 

  $A\cap B or Ac\cap Bc$

Also these events will be mutually exclusive :

  $P(A\cap B)+P(Ac\cap Bc)=P\left(A\right)P\left(B\right)P\left(C\right)=\frac{2}{3}*\frac{4}{5}+\frac{1}{3}*\frac{1}{5}=\frac{3}{5}$