grandes-ecoles 2019 Q8

grandes-ecoles · France · centrale-maths1__psi Discrete Probability Distributions Recurrence Relations and Sequences Involving Probabilities
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
  • we randomly draw a ball from the urn;
  • we replace the drawn ball in the urn;
  • we add to the urn a ball of the same color as the drawn ball.
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws.
Let $n$ and $k$ be two integers greater than or equal to 1. Establish that $$P(X_{n} = k) = \frac{k-1}{n+1} P(X_{n-1} = k-1) + \frac{n+1-k}{n+1} P(X_{n-1} = k).$$ 
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
\begin{itemize}
  \item we randomly draw a ball from the urn;
  \item we replace the drawn ball in the urn;
  \item we add to the urn a ball of the same color as the drawn ball.
\end{itemize}
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws.

Let $n$ and $k$ be two integers greater than or equal to 1. Establish that
$$P(X_{n} = k) = \frac{k-1}{n+1} P(X_{n-1} = k-1) + \frac{n+1-k}{n+1} P(X_{n-1} = k).$$
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