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Let the independent random variables $X_1$ and $X_2$ have a binomial distribution with parameters $n_1 = 3$, $p_1 = 2/3$ and $n_2 = 4$, $p_2 = 1/2$, respectively. Compute $P(X_1 = X_2)$.

Hint: List the four mutually exclusive ways that $X_1 = X_2$ and compute the probability of each.

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What have you tried? –  Stefan Hansen Nov 7 '12 at 11:55
I don't know where to start from. I know the formula for binomial (n,p) Both X1 and X1 will have four mutually exclusive ways when n=0, 1, 2, 3. –  user48495 Nov 7 '12 at 14:09
$X_1$ can attain the values $0,1,2,3$ (since $n_1=3$) and $X_2$ can attain the values $0,1,2,3,4$ (since $n_2=4$). This means that $X_1=X_2$ can only happen when they both equal to either $0,1,2,3$, i.e. $\{X_1=X_2\}=\bigcup_{i=0}^3 \{X_1=X_2=i\}$. Does this help? –  Stefan Hansen Nov 7 '12 at 14:17
We can write the union as $$\{X_1=X_2\}=\bigcup_{i=0}^3 (\{X_1=i\}\cap \{X_2=i\}).$$ Now, because they are mutually exclusive we have that $$P(X_1=X_2)=\sum_{i=0}^4 P(\{X_1=i\}\cap \{X_2=i\}).$$ Now use that they are independent. –  Stefan Hansen Nov 7 '12 at 14:42
You should post a new question with this :) –  Stefan Hansen Nov 7 '12 at 15:14

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