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Fie \( X_{1},X_{2},...,X_{n} \) variabile aleatoare independente cu repartitiile \( P(X_{i}=k)=pq^{k},i=1,2;k=0,1,2,3... \).Sa se arate ca \( P(X_{1}=k/X_{1}+X_{2}=n)= \frac{1}{n+1},k=1,n. \)

Se aplica formula probabilitatilor conditionate:
\( P(X_1=k | X_1+X_2=n )=\frac{P(X_1=k)P(X_2=n-k)}{P(X_1+X_2=n)}=\frac{pq^kpq^{n-k}}{\sum_{i=0}^n pq^ipq^{n-i}}=\frac{1}{n+1} \).