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Determine the sample space for an experiment if a coin is selected from a bag three times. There are three coins in the bag: a nickel (N), a quarter (Q), and a dime (D). Each time a coin is selected, it is replaced before selecting the next coin. (1 point)

NNN, NNQ, NND, NQN, NQD, NQQ, NDN, NDQ, NDD, QQQ, QQD, QQN,
QDQ, QDN, QDD, QNQ, QND, QNN
NNN, NNQ, NND, NQN, NQD, NQQ, NDN, NDQ, NDD, DDD, DDQ, DDN,
O DQD, DQN, DQQ, DND, DNQ, DNN, QQQ, QQD, QQN, QDQ, QDN, QDD,
QNQ, QND, QNN
NNN, NNQ, NND, NQN, NQD, NQQ, NDN, NDQ, NDD, NNN, DDQ, DDN,
O DQD, DQN, DQQ, DND, DNQ, DNN, DDD, QQD, QQN, QDQ, QDN, QDD,
QNQ, QND, QNN
NNN, NNQ, NND, NQN, NQD, NQQ, NDN, NDQ, NDD, DDD, DDQ, DDN,
DQD, DQN, DQQ, DND, DNQ, DNN

1 answer

To determine the sample space for the experiment of selecting a coin from a bag three times (with replacement) when the bag contains three types of coins (a nickel (N), a quarter (Q), and a dime (D)), we consider all possible outcomes.

Each selection can result in one of three coins (N, Q, or D), and since the selection is made three times, we can represent each outcome as a sequence of three letters, where each letter corresponds to the coin selected on that draw.

The total number of outcomes can be calculated as:

  • For the first selection, you have 3 options (N, Q, D),
  • For the second selection, you also have 3 options (N, Q, D),
  • For the third selection, you once again have 3 options (N, Q, D).

Thus, the total number of outcomes is \(3 \times 3 \times 3 = 3^3 = 27\).

Now, we can list all the possible outcomes, which make up the sample space:

  1. NNN
  2. NNQ
  3. NND
  4. NQN
  5. NQQ
  6. NQD
  7. NDN
  8. NDQ
  9. NDD
  10. QNN
  11. QNQ
  12. QQN
  13. QQQ
  14. QQD
  15. QDQ
  16. QDN
  17. QDD
  18. DNN
  19. DNQ
  20. DQN
  21. DQQ
  22. DQD
  23. DDN
  24. DDD
  25. DDQ
  26. DND
  27. DNQ

So, the correct sample space is: NNN, NNQ, NND, NQN, NQQ, NQD, NDN, NDQ, NDD, QNN, QNQ, QQN, QQQ, QQD, QDQ, QDN, QDD, DNN, DNQ, DQN, DQQ, DQD, DDN, DDD, DDQ, DND, DNQ

Which can be summarized, confirming the previous exploration:

  1. NNN
  2. NNQ
  3. NND
  4. NQN
  5. NQQ
  6. NQD
  7. NDN
  8. NDQ
  9. NDD
  10. QNN
  11. QNQ
  12. QQN
  13. QQQ
  14. QQD
  15. QDQ
  16. QDN
  17. QDD
  18. DNN
  19. DNQ
  20. DQN
  21. DQQ
  22. DQD
  23. DDN
  24. DDD
  25. DDQ
  26. DND
  27. DNQ

This shows all \(27\) outcomes in the complete sample space.