In the example shown below, we consider that expected payoff of both the row's action will be the same. While calculating the expected payoff of the action we use the probability of the column, shouldn't we also consider the probability of the row action, whose payoff we are calculating?
In equilibrium, every player must play a best response. Randomising between two pure strategies is a best response only if the player is indifferent between them.
Why? Think of it as tossing a coin to decide what to do as Player 1. If you like B strictly more than F, tossing a coin is not a best response because you might end up with F and miss out on B (which you like better).. But if you are indifferent between B and F, whatever the coin says you end up with the same utility.
Precisely because you are indifferent between B and F, you are also indifferent whether the coin is fair or biased. Even if B is more likely to be chosen, you always end up with the same utility. So the probability of the row action affects the likelihood that you play B or F, but it is irrelevant for your utility.
On the other hand, the probabilities of the column action must be finely balanced to make sure that you are indifferent.