The problem is as follows:
Draw cards at random with replacement from a standard deck of cards. Given that face cards are Jack, Queen and King, what is the probability that the 3rd face card is drawn on the 8th draw?
The correct answer given is ${7 \choose 2}(\frac{3}{13} )^3(\frac{10}{13})^5 $ But I don't understand why taking the probability of getting 3 successes out of 8 which is ${8 \choose 3}(\frac{3}{13} )^3(\frac{10}{13})^5$ And subtracting the probability of having 3 successes in seven ${7 \choose 3}(\frac{3}{13} )^3(\frac{10}{13})^4$ Yielding ${8 \choose 3}(\frac{3}{13} )^3(\frac{10}{13})^5-{7 \choose 3}(\frac{3}{13} )^3(\frac{10}{13})^4$ is incorrect. It makes sense to me intuitively. Can someone help me understand this please?
The error in your computation is in the term you are subtracting.
The first term in your difference correctly computes the probability that you will get exactly three face cards in the first eight draws. Now, you want to subtract those cases in which the eighth draw is a non-face card. Thus you need exactly three face cards out of the first seven, and you need the eight draw to be a non-face card. Thus, your expression needs an extra factor of $\frac {10}{13}$.