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

1 answer

When selecting a coin from a bag three times, with replacement, each selection can be one of three coins: a nickel (N), a quarter (Q), or a dime (D). The total number of outcomes is determined by the number of choices (3 coins) raised to the number of selections (3 times).

Let's denote the three coins:

  • N = nickel
  • Q = quarter
  • D = dime

The sample space consists of all possible outcomes of the experiment of selecting a coin three times. Since each selection is independent (because of replacement), we can calculate the sample space outcomes as follows:

For each of the three selections:

  1. The first selection has 3 possible outcomes (N, Q, D).
  2. The second selection also has 3 possible outcomes (N, Q, D).
  3. The third selection has 3 possible outcomes (N, Q, D).

Therefore, the total number of combinations is: \[ 3 \times 3 \times 3 = 27 \]

Now we can enumerate all the combinations:

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

Arranging these systematically leads to the complete sample space:

Thus, the sample space is: \[ { NNN, NNQ, NND, NQN, NQQ, NQD, NDN, NDQ, NDD, DNN, DNQ, DND, DQN, DQQ, DDD, QNN, QNQ, QND, QQN, QQD, QDQ, QDN, QQQ, QDD } \]

This should include all possible outcomes from selecting three coins with replacement.