<Dd> Pr (Y = m) = ∑ k = m n Pr (Y = m ∣ X = k) Pr (X = k) = ∑ k = m n (n k) (k m) p k q m (1 − p) n − k (1 − q) k − m (\ displaystyle (\ begin (aligned) \ Pr (Y = m) & = \ sum _ (k = m) ^ (n) \ Pr (Y = m \ mid X = k) \ Pr (X = k) \ \ (2pt) & = \ sum _ (k = m) ^ (n) (\ binom (n) (k)) (\ binom (k) (m)) p ^ (k) q ^ (m) (1 - p) ^ (n-k) (1 - q) ^ (k-m) \ \ \ end (aligned))) </Dd> <P> Since (n k) (k m) = (n m) (n − m k − m) (\ displaystyle \ scriptstyle (\ binom (n) (k)) (\ binom (k) (m)) = (\ binom (n) (m)) (\ binom (n-m) (k-m))), the equation above can be expressed as </P> <Dl> <Dd> Pr (Y = m) = ∑ k = m n (n m) (n − m k − m) p k q m (1 − p) n − k (1 − q) k − m (\ displaystyle \ Pr (Y = m) = \ sum _ (k = m) ^ (n) (\ binom (n) (m)) (\ binom (n-m) (k-m)) p ^ (k) q ^ (m) (1 - p) ^ (n-k) (1 - q) ^ (k-m)) </Dd> </Dl> <Dd> Pr (Y = m) = ∑ k = m n (n m) (n − m k − m) p k q m (1 − p) n − k (1 − q) k − m (\ displaystyle \ Pr (Y = m) = \ sum _ (k = m) ^ (n) (\ binom (n) (m)) (\ binom (n-m) (k-m)) p ^ (k) q ^ (m) (1 - p) ^ (n-k) (1 - q) ^ (k-m)) </Dd>

What is the definition of success in this binomial distribution application