A box contains one coupon labelled $1$, two coupons labelled $2$, and so on up to ten coupons labelled $10$. Two distinct coupons are drawn at random from the box. Given that the two coupons carry the same label, find the conditional probability that the common label is $k$ (where $k$ is $1$ or $2$ or $... 10$)
Any hints? Total number of coupons we have is $5\cdot 11 = 55$.
On
The total number of choices for two coupons with the same label is
$\left(\begin{matrix} 2 \\ 2\end{matrix}\right)+\left(\begin{matrix} 3 \\ 2\end{matrix}\right)+ ... +\left(\begin{matrix} 10 \\ 2\end{matrix}\right)=\frac {9.10.11}{6}=165.$
So the conditional probability both labels are $k$ is $$\frac{1}{165}\left(\begin{matrix} k \\ 2\end{matrix}\right).$$
Hint:
Let $X_1$ denote the label of the coupon drawn firstly and $X_2$ the label of the coupon drawn secondly.
We have:
$$\begin{aligned}P\left(X_{1}=k\mid X_{1}=X_{2}\right)P\left(X_{1}=X_{2}\right) & =P\left(X_{1}=X_{2}=k\right)\\ & =P\left(X_{1}=k\right)P\left(X_{2}=k\mid X_{1}=k\right) \end{aligned} \tag1$$
Now find $P(X_1=X_2), P(X_1=k)$ and $P(X_2=k\mid X_1=k)$.
The hardest to find is $P(X_1=X_2)$ but for that you can make use of:$$P(X_1=X_2)=\sum_{i=2}^{10}P(X_1=i=X_2)=\sum_{i=2}^{10}P(X_1=i)P(X_2=i\mid X_1=i)$$
Then $(1)$ can be used in order to find $P\left(X_{1}=k\mid X_{1}=X_{2}\right)$.