Question

abel is playing a game with a standard deck of cards and randomly chooses a spade on his first draw.

which of the following statements is true?

• if abel draws a spade, replaces the card, re-shuffles, and draws again, these will be independent events.
• if abel draws a spade, does not replace the card, and draws again, these will be independent events.
• whether or not these are dependent or independent events depends on what type of card abel draws next.
• if abel draws a spade, replaces the card, re-shuffles, and draws again, these will be dependent events.

Explanation:

Analyze the concept of independent events

Using the Independent Events Probability knowledge point

Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other. In card games, drawing with replacement (putting the card back and shuffling) preserves the composition of the deck, meaning the probability of drawing any suit on the second draw remains exactly the same as on the first draw.

Analyze the concept of dependent events

Using the Dependent Events Probability knowledge point

If a card is drawn and not replaced, the total number of cards and the number of cards of that specific suit in the deck change. This alters the probability of the subsequent draw, making the two draws dependent events.

Evaluate the given statements

• Statement 1: "If Abel draws a spade, replaces the card, re-shuffles, and draws again, these will be independent events."
• Since the card is replaced and the deck is re-shuffled, the probability of drawing a spade on the second draw remains \(\frac{13}{52} = \frac{1}{4}\). The outcome of the first draw does not affect the second draw. This statement is true.
• Statement 2: "If Abel draws a spade, does not replace the card, and draws again, these will be independent events."
• Without replacement, the probability of drawing a spade on the second draw changes to \(\frac{12}{51}\). The events are dependent, so this statement is false.
• Statement 3: "Whether or not these are dependent or independent events depends on what type of card Abel draws next."
• Independence depends on the process (replacement vs. non-replacement), not the specific outcome of the next draw. This statement is false.
• Statement 4: "If Abel draws a spade, replaces the card, re-shuffles, and draws again, these will be dependent events."
• As established, replacement makes them independent, so this statement is false.

Answer:

• (A) If Abel draws a spade, replaces the card, re-shuffles, and draws again, these will be independent events. (Correct answer)
• (B) If Abel draws a spade, does not replace the card, and draws again, these will be independent events.
• (C) Whether or not these are dependent or independent events depends on what type of card Abel draws next.
• (D) If Abel draws a spade, replaces the card, re-shuffles, and draws again, these will be dependent events.