To construct a probability model for the experiment of picking a card from a selection of cards numbered 5, 10, and 15, we first define the random variable \( X \) which represents the value of the card picked.
The possible values of \( X \) are:
- \( 5 \)
- \( 10 \)
- \( 15 \)
In ascending order, the values of \( X \) are: \[ X = 5, 10, 15 \]
Next, let's determine the probability model \( f(x) \). Assuming that each card has an equal chance of being selected, the probabilities for each value of \( X \) can be calculated. Since there are 3 cards, and each card is picked with equal probability, the probability for each value is:
- \( P(X = 5) = \frac{1}{3} \)
- \( P(X = 10) = \frac{1}{3} \)
- \( P(X = 15) = \frac{1}{3} \)
Thus, the probability model \( f(x) \) is given by:
\[ f(x) = \begin{cases} \frac{1}{3}, & \text{if } x = 5 \ \frac{1}{3}, & \text{if } x = 10 \ \frac{1}{3}, & \text{if } x = 15 \ 0, & \text{otherwise} \end{cases} \]
In a summarized form, we can write:
- \( f(x) = \frac{1}{3} \) for \( x = 5, 10, 15 \)
So the final answer for the probability model and the values of \( X \) is:
\[ f(x) = \begin{cases} \frac{1}{3} & \text{for } x = 5 \ \frac{1}{3} & \text{for } x = 10 \ \frac{1}{3} & \text{for } x = 15 \ 0 & \text{otherwise} \end{cases} \]
where \( x = 5, 10, 15 \).