5. find the probability that when rolling a standard die 4 times, an odd number only shows up twice.

• k}=(\frac{1}{2})^{2}$, so multiplying them gives $(\frac{1}{2})^{2+2}=(\frac{1}{2})^{4}=\frac{1}{16}$? Wait, no, $(\frac{1}{2})^{2}\times(\frac{1}{2})^{2}=\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}$? Wait, no, $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$, so $(\frac{1}{4})\times(\frac{1}{4})=\frac{1}{16}$? Wait, no, $p^{k}=(\frac{1}{2})^{2}=\frac{1}{4}$, $(1 - p)^{n - k}=(\frac{1}{2})^{2}=\frac{1}{4}$, then $\binom{4}{2}\times\frac{1}{4}\times\frac{1}{4}=6\times\frac{1}{16}=\frac{6}{16}=\frac{3}{8}$. Wait, let's recalculate: $\binom{4}{2}=6$, $p^{k}=(\frac{1}{2})^{2}=\frac{1}{4}$, $(1 - p)^{n - k}=(\frac{1}{2})^{2}=\frac{1}{4}$. Then $P(X = 2)=6\times\frac{1}{4}\times\frac{1}{4}=6\times\frac{1}{16}=\frac{6}{16}=\frac{3}{8}$. Wait, no, $\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}$, 6 times $\frac{1}{16}$ is $\frac{6}{16}=\frac{3}{8}$. Wait, actually, $(\frac{1}{2})^{2}\times(\frac{1}{2})^{2}=(\frac{1}{2})^{4}=\frac{1}{16}$? No, $(\frac{1}{2})^{2}$ is $\frac{1}{4}$, $(\frac{1}{2})^{2}$ is $\frac{1}{4}$, multiplying them gives $\frac{1}{16}$? Wait, no, $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$, so $(\frac{1}{2})^{2}=\frac{1}{4}$, $(\frac{1}{2})^{2}=\frac{1}{4}$, so $\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}$? Wait, no, that's wrong. $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$, and then $\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}$? Wait, no, if you have $(\frac{1}{2})^{2}\times(\frac{1}{2})^{2}$, that's $(\frac{1}{2})^{2 + 2}=(\frac{1}{2})^{4}=\frac{1}{16}$. But 6 times $\frac{1}{16}$ is $\frac{6}{16}=\frac{3}{8}$. Wait, let's check with another approach. The number of ways to choose 2 positions out of 4 for the odd numbers: $\binom{4}{2}=6$. For each of these ways, the probability of getting an odd number in those 2 positions is $(\frac{1}{2})^{2}$, and even in the other 2 positions is $(\frac{1}{2})^{2}$. So total probability is $6\times(\frac{1}{2})^{2}\times(\frac{1}{2})^{2}=6\times\frac{1}{4}\times\frac{1}{4}=6\times\frac{1}{16}=\frac{6}{16}=\frac{3}{8}$.

Answer:

$\frac{3}{8}$