Question
A deck of 40 cards contains 20 blue and 20 red cards. Each color has cards numbered 1 to 20.
Which events are independent? Choose ,begin emphasis,all,end emphasis, that apply.
Answer options with 6 options
A.
A 2 is chosen from the deck. The card is put back into the deck and then a 2 is chosen.
B.
A 2 is chosen from the deck. The card is not put back into the deck and then a 2 is chosen.
C.
A red 5 is chosen from the deck. The card is put back into the deck and then a red card is chosen.
D.
A red 5 is chosen from the deck. The card is not put back into the deck and then a red card is chosen.
E.
A red card is chosen from the deck. The card is put back into the deck and then a blue card is chosen.
F.
A red card is chosen from the deck. The card is not put back into the deck and then a blue card is chosen.
Which events are independent? Choose ,begin emphasis,all,end emphasis, that apply.
Answer options with 6 options
A.
A 2 is chosen from the deck. The card is put back into the deck and then a 2 is chosen.
B.
A 2 is chosen from the deck. The card is not put back into the deck and then a 2 is chosen.
C.
A red 5 is chosen from the deck. The card is put back into the deck and then a red card is chosen.
D.
A red 5 is chosen from the deck. The card is not put back into the deck and then a red card is chosen.
E.
A red card is chosen from the deck. The card is put back into the deck and then a blue card is chosen.
F.
A red card is chosen from the deck. The card is not put back into the deck and then a blue card is chosen.
Answers
GPT-4o mini
To determine which events are independent, we need to assess whether the outcome of the first event affects the outcome of the second event.
A. **A 2 is chosen from the deck. The card is put back into the deck and then a 2 is chosen.**
- These events are independent because replacing the card ensures that the probability of drawing a 2 remains the same on both draws.
B. **A 2 is chosen from the deck. The card is not put back into the deck and then a 2 is chosen.**
- These events are NOT independent because not replacing the card changes the probability of drawing a second 2. After removing one 2 (if it was chosen), there is only one 2 left out of 39 cards.
C. **A red 5 is chosen from the deck. The card is put back into the deck and then a red card is chosen.**
- These events are independent because replacing the card ensures that the probability of drawing a red card in the second event is not affected by the first event.
D. **A red 5 is chosen from the deck. The card is not put back into the deck and then a red card is chosen.**
- These events are NOT independent because not replacing the card after drawing the red 5 may affect the number of red cards left and therefore the probability of drawing another red card in the second event.
E. **A red card is chosen from the deck. The card is put back into the deck and then a blue card is chosen.**
- These events are independent because replacing the card ensures that the outcome of the first event does not affect the probability of drawing a blue card afterwards.
F. **A red card is chosen from the deck. The card is not put back into the deck and then a blue card is chosen.**
- These events are NOT independent because choosing a red card and not replacing it reduces the total number of cards available in the deck, potentially affecting the probability of drawing a blue card in the second event.
Thus, the independent events are:
- A (2 is chosen and replaced)
- C (red 5 chosen and replaced)
- E (red card chosen and replaced)
So the final answer is **A, C, and E.**
A. **A 2 is chosen from the deck. The card is put back into the deck and then a 2 is chosen.**
- These events are independent because replacing the card ensures that the probability of drawing a 2 remains the same on both draws.
B. **A 2 is chosen from the deck. The card is not put back into the deck and then a 2 is chosen.**
- These events are NOT independent because not replacing the card changes the probability of drawing a second 2. After removing one 2 (if it was chosen), there is only one 2 left out of 39 cards.
C. **A red 5 is chosen from the deck. The card is put back into the deck and then a red card is chosen.**
- These events are independent because replacing the card ensures that the probability of drawing a red card in the second event is not affected by the first event.
D. **A red 5 is chosen from the deck. The card is not put back into the deck and then a red card is chosen.**
- These events are NOT independent because not replacing the card after drawing the red 5 may affect the number of red cards left and therefore the probability of drawing another red card in the second event.
E. **A red card is chosen from the deck. The card is put back into the deck and then a blue card is chosen.**
- These events are independent because replacing the card ensures that the outcome of the first event does not affect the probability of drawing a blue card afterwards.
F. **A red card is chosen from the deck. The card is not put back into the deck and then a blue card is chosen.**
- These events are NOT independent because choosing a red card and not replacing it reduces the total number of cards available in the deck, potentially affecting the probability of drawing a blue card in the second event.
Thus, the independent events are:
- A (2 is chosen and replaced)
- C (red 5 chosen and replaced)
- E (red card chosen and replaced)
So the final answer is **A, C, and E.**