/raffle

a very hard probability question

Primary LanguageC++

Problem statement

The Yankees are holding a 50/50 raffle that works as follows:

  • The patrons will buy entries for a certain amount of money. All of the money is collected together.
  • A single ticket is chosen at random from the submitted entries of all the patrons.
  • The selected winner gets 50% of the prize money, and the other 50% is donated to charity. Let $n$ be the total number of patrons that partake in the raffle.

Patrons can buy entries using one of the four configurations:

  1. 5 entries for 10 USD ($\mathcal{C}_1$).
  2. 20 entries for 20 USD ($\mathcal{C}_2$).
  3. 100 entries for 40 USD ($\mathcal{C}_3$).
  4. 300 entries for 100 USD ($\mathcal{C}_4$).

We will call each of these four groups a cohort.

Suppose also that patrons have the following probabilities of belonging to a given cohort:

  1. 15% are in $\mathcal{C}_1$.
  2. 10% are in $\mathcal{C}_2$.
  3. 30% are in $\mathcal{C}_3$.
  4. 45% are in $\mathcal{C}_4$.

Given these probabilities, we have the two following questions:

  1. what is the expected value of the prize money?
  2. what is the expected value of a patron's earnings for the raffle?

Assumptions

Even though it is not stated, we make the following assumptions about the raffle:

  • A patron can only belong to a single cohort.
  • A patron can only buy a single set of entries. i.e. if they are in $\mathcal{C}_2$, they will only contribute 20 entries and 20 USD total.

Definitions and notation

Let $n$ be the number of patrons that participate in the raffle.

For a given cohort $\mathcal{C}_k$, let:

  • $e_k$ be the number of entries submitted by a patron in that cohort ($e_1 = 5$, $e_2 = 20$, $e_3 = 100$, $e_4 = 300$).
  • $v_k$ be the value per entry, which is just the total value divided by the number of entries (e.g. $v_1 = 10 / 5 = 2$). So $v_1 = 2$, $v_2 = 1$, $v_3 = 2 / 5$, and $v_4 = 1 / 3$.

We define the following:

  • $\mathbf{p} = [0.15, 0.10, 0.30, 0.45]^\top$ is the vector of probabilities of belonging to a given cohort.

  • $\mathbf{N} = [N_1, N_2, N_3, N_4]^\top$ is the number of patrons that belong to each cohort. Given $n$ and $\mathbf{p}$, we can see $\mathbf{N}$ is a multinomial random variable, i.e.

$$\mathbf{N} \sim \mathrm{MN}(n, \mathbf{p}) \text{, with density } f_{\mathbf{N}}(\mathbf{n}) = \frac{n!}{n_1 ! \cdot n_2 ! \cdot n_3 ! \cdot n_4 !} \, p_1^{n_1} p_2^{n_2} p_3^{n_3} p_4^{n_4}.$$

  • It's worth noting that for each cohort, we have $\mathbb{E}[N_k] = n p_k$ and $\mathbb{V}\mathrm{ar}[N_k] = n p_k (1 - p_k)$.

  • $\mathcal{P}_n$ be the set of all $\mathbf{n}$ such that $\mathbf{1}^\top\mathbf{n} = n$, i.e. the domain of $\mathbf{N}$.

  • $\mathbf{E} = [E_1, E_2, E_3, E_4]^\top$ is the number of entries submitted by each cohort. For each cohort we have $$E_k = e_k \cdot N_k.$$

  • $E$ is the total number of entries. This is just

$$E = \mathbf{1}^\top\mathbf{E} = \sum_{k = 1}^4 e_k N_k.$$

  • $\mathbf{V} = [V_1, V_2, V_3, V_4]^\top$ is the total value from each cohort. For each cohort we have $$V_k = v_k \cdot E_k = v_k \cdot e_k \cdot N_k.$$

  • $V$ is the total value, just

$$V = \mathbf{1}^\top\mathbf{V} = \sum_{k = 1}^4 v_k E_k = \sum_{k = 1}^4 v_k e_k N_k.$$
  • $T$ is the patron value, just $V / 2$.

From these definitions, we see that both the entry random variables ($\mathbf{E}$ and $E$) and the value random variables ($\mathbf{N}$ and $V$) depend on $\mathbf{N}$. Thus these are really just functions of the random variable $\mathbf{N}$, which is how we will go about solving the problem.

Revisiting first problem

The first problem now amounts to determining $\mathbb{E} \left[ T \right]$. Expanding out $T$ gives us

$$T(\mathbf{N}) = \frac{1}{2} V(\mathbf{N}) = \frac{1}{2} \sum_{k = 1}^4 v_k e_k N_k.$$

The naive approach would be to compute

$$\mathbb{E}[T] = \mathbb{E} \left[ \frac{1}{2} \sum_{k = 1}^4 v_k e_k N_k \right] = \frac{1}{2} \sum_{k = 1}^4 v_k e_k \mathbb{E}[N_k] = \frac{n}{2} \sum_{k = 1}^4 v_k e_k p_k,$$

but the random variables $\mathbf{N}$ are not independent.

Instead, we should compute

$$\mathbb{E}[T] = \sum_{\mathbf{n} \in \mathcal{P}_n} T(\mathbf{n}) \cdot f(\mathbf{n}), = \frac{1}{2} \sum_{\mathbf{n} \in \mathcal{P}_n} f(\mathbf{n}) \sum_{k = 1}^4 v_k e_k n_k,$$

i.e. we would have to sum over all plausible combinations of cohort.

Revisiting second problem