Question

Jon chooses one of these cards. Then, he shuffles the card back into the deck and picks a card again.

Which list represents the sample space of this compound event?

Answers

GPT-4o mini
To create a sample space for the compound event described, we need to consider that Jon is picking a card from a deck, replacing it back into the deck (shuffling), and picking a card again. This means that every card he could pick in the first draw could be picked again in the second draw.

Assuming we are working with a standard deck of 52 cards, we would represent the sample space of the compound event (picking a card, putting it back, and picking another card) as all possible pairs of cards that could be drawn.

For example, if we denote the cards in the deck simply (without suits) as:

- 1, 2, 3, ..., 52

The sample space would include pairs like:

- (1, 1), (1, 2), (1, 3), ..., (1, 52)
- (2, 1), (2, 2), (2, 3), ..., (2, 52)
- ...
- (52, 1), (52, 2), (52, 3), ..., (52, 52)

Since there are 52 possibilities for the first card and 52 possibilities for the second card, the total number of outcomes in the sample space is 52 x 52 = 2704 possible pairs.

In general notation, the sample space \( S \) can be described as:

\[ S = \{ (i, j) \: | \: i, j \in \{1, 2, \ldots, 52\} \} \]

This means the sample space includes every combination of two draws from the deck, considering each draw is independent of the other due to replacement.

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