Question

Let a card be selected from an ordinary deck of playing cards. The outcome $c$ is one of these 52 cards. Let $X(c)=4$ if $c$ is an ace, let $X(c)=3$ if $c$ is a king, let $X(c)=2$ if $c$ is a queen, let $X(c)=1$ if $c$ is a jack, and let $X(c)=0$ otherwise. Suppose that $P$ assigns a probability of $\frac{1}{52}$ to each outcome $c.$ Describe the induced probability $P_X(D)$ on the space $$\begin{gathered} \mathcal{D}= \\ 0,1,2,3,4 \end{gathered}$$ of the random variable $X$.

Short answer

The induced probability $P_X(D)$ on the space $\mathcal{D}=0,1,2,3,4$ of the random variable $X$ is: $P_X(4)=P_X(3)=P_X(2)=P_X(1)=4\cdot \frac{1}{52}=\frac{1}{13}$ and $P_X(0)=36\cdot \frac{1}{52}=\frac{9}{13}$.

Step-by-step solution

Understand the Random Variable Definition

In this exercise, a random variable $X$ is defined for the outcomes $c$ as follows: $X(c)=4$ if $c$ is an ace, $X(c)=3$ if $c$ is a king, $X(c)=2$ if $c$ is a queen, $X(c)=1$ if $c$ is a jack, and $X(c)=0$ otherwise. The total possibilities are 52 since we have a standard deck of cards.

Enumerate the Events for Each Value of the Random Variable

With the deck of 52 cards, we have 4 Aces, 4 Kings, 4 Queens, and 4 Jacks. The rest of the cards are neither an ace, king, queen, nor jack. So when $X(c)=0$, that means the card selected is not an Ace, King, Queen or Jack. That leaves us with $52-4-4-4-4=36$ such cards.

Compute Probabilities for Each Value of the Random Variable

The probability for each outcome $c$ is given as $\frac{1}{52}$. So, for each specific value of the random variable $X(c)$: \n\n1. The probability $P_X(4)$ of getting an Ace (4 Aces in a deck) is $4\cdot \frac{1}{52}$.\n\n2. The probability $P_X(3)$ of getting a King (4 Kings in a deck) is $4\cdot \frac{1}{52}$. \n\n3. The probability $P_X(2)$ of getting a Queen (4 Queens in a deck) is $4\cdot \frac{1}{52}$. \n\n4. The probability $P_X(1)$ of getting a Jack (4 Jacks in a deck) is $4\cdot \frac{1}{52}$. \n\n5. Finally, the probability $P_X(0)$ of getting any other card (36 such cards in a deck) is $36\cdot \frac{1}{52}$.