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Sagot :
Let's take a look at the problem step by step to determine which list represents the sample space for pulling 2 marbles from the bag with replacement.
Given lists are:
- List 1: [tex]\( G_1, G_2, Y, P \)[/tex]
- List 2: [tex]\( G_1, G_1, G_2, G_1, Y, G_1, P, G_2, G_1, G_2, Y, G_2, P, Y, G_1, Y, G_2, Y, P, P, G_1, P, G_2, P, Y \)[/tex]
- List 3: [tex]\( G_1, G_1, G_1, G_1, G_1, G_2, G_1, G_2, G_1, Y, G_1, Y, G_1, P, G_1, P, G_2, G_1, G_2, G_2, G_2, Y, G_2, Y, G_2, P, G_2, P, Y, G_1, Y, G_2, Y, Y, Y, P, Y, P, G_1, P, G_2, P, Y, P, P, P, P \)[/tex]
- List 4: [tex]\( G_1, G_1, G_1, G_1, G_1, G_2, G_1, G_2, G_1, Y, G_1, Y, G_1, P, G_1, P, G_2, G_1, G_2, G_2, G_2, Y, G_2, Y, G_2, P, G_2, P, Y, G_1, Y, G_2, Y, Y, Y, P, Y, P, G_1, P, G_2, P, Y, P, P, P, P \)[/tex]
For pulling 2 marbles from the bag with replacement, each marble drawing is independent, and we consider all combinations of the marbles. The sample space for this scenario includes all possible ordered pairs of marbles.
Given the set of marbles in the bag: [tex]\( G_1, G_2, Y, P \)[/tex], we can form pairs by drawing one marble and then drawing another one with replacement.
Thus, the sample space is composed of the following pairs:
- When the first marble is [tex]\( G_1 \)[/tex]: [tex]\( (G_1, G_1), (G_1, G_2), (G_1, Y), (G_1, P) \)[/tex]
- When the first marble is [tex]\( G_2 \)[/tex]: [tex]\( (G_2, G_1), (G_2, G_2), (G_2, Y), (G_2, P) \)[/tex]
- When the first marble is [tex]\( Y \)[/tex]: [tex]\( (Y, G_1), (Y, G_2), (Y, Y), (Y, P) \)[/tex]
- When the first marble is [tex]\( P \)[/tex]: [tex]\( (P, G_1), (P, G_2), (P, Y), (P, P) \)[/tex].
Aggregating all these pairs, we get:
[tex]\[ \{(G_1, G_1), (G_1, G_2), (G_1, Y), (G_1, P), (G_2, G_1), (G_2, G_2), (G_2, Y), (G_2, P), (Y, G_1), (Y, G_2), (Y, Y), (Y, P), (P, G_1), (P, G_2), (P, Y), (P, P)\} \][/tex].
This matches the result:
[tex]\[ ('G_1', 'G_1'), ('G_1', 'G_2'), ('G_1', 'Y'), ('G_1', 'P'), ('G_2', 'G_1'), ('G_2', 'G_2'), ('G_2', 'Y'), ('G_2', 'P'), ('Y', 'G_1'), ('Y', 'G_2'), ('Y', 'Y'), ('Y', 'P'), ('P', 'G_1'), ('P', 'G_2'), ('P', 'Y'), ('P', 'P') \][/tex].
In these lists:
- List 1 only lists individual marbles.
- List 2 lists combinations but not considering replacement.
- List 3 lists the correct pairs but in a nested format.
- List 4 accurately represents the sample space as described above.
Hence, List 4 correctly provides the sample space for pulling 2 marbles from the bag with replacement.
Given lists are:
- List 1: [tex]\( G_1, G_2, Y, P \)[/tex]
- List 2: [tex]\( G_1, G_1, G_2, G_1, Y, G_1, P, G_2, G_1, G_2, Y, G_2, P, Y, G_1, Y, G_2, Y, P, P, G_1, P, G_2, P, Y \)[/tex]
- List 3: [tex]\( G_1, G_1, G_1, G_1, G_1, G_2, G_1, G_2, G_1, Y, G_1, Y, G_1, P, G_1, P, G_2, G_1, G_2, G_2, G_2, Y, G_2, Y, G_2, P, G_2, P, Y, G_1, Y, G_2, Y, Y, Y, P, Y, P, G_1, P, G_2, P, Y, P, P, P, P \)[/tex]
- List 4: [tex]\( G_1, G_1, G_1, G_1, G_1, G_2, G_1, G_2, G_1, Y, G_1, Y, G_1, P, G_1, P, G_2, G_1, G_2, G_2, G_2, Y, G_2, Y, G_2, P, G_2, P, Y, G_1, Y, G_2, Y, Y, Y, P, Y, P, G_1, P, G_2, P, Y, P, P, P, P \)[/tex]
For pulling 2 marbles from the bag with replacement, each marble drawing is independent, and we consider all combinations of the marbles. The sample space for this scenario includes all possible ordered pairs of marbles.
Given the set of marbles in the bag: [tex]\( G_1, G_2, Y, P \)[/tex], we can form pairs by drawing one marble and then drawing another one with replacement.
Thus, the sample space is composed of the following pairs:
- When the first marble is [tex]\( G_1 \)[/tex]: [tex]\( (G_1, G_1), (G_1, G_2), (G_1, Y), (G_1, P) \)[/tex]
- When the first marble is [tex]\( G_2 \)[/tex]: [tex]\( (G_2, G_1), (G_2, G_2), (G_2, Y), (G_2, P) \)[/tex]
- When the first marble is [tex]\( Y \)[/tex]: [tex]\( (Y, G_1), (Y, G_2), (Y, Y), (Y, P) \)[/tex]
- When the first marble is [tex]\( P \)[/tex]: [tex]\( (P, G_1), (P, G_2), (P, Y), (P, P) \)[/tex].
Aggregating all these pairs, we get:
[tex]\[ \{(G_1, G_1), (G_1, G_2), (G_1, Y), (G_1, P), (G_2, G_1), (G_2, G_2), (G_2, Y), (G_2, P), (Y, G_1), (Y, G_2), (Y, Y), (Y, P), (P, G_1), (P, G_2), (P, Y), (P, P)\} \][/tex].
This matches the result:
[tex]\[ ('G_1', 'G_1'), ('G_1', 'G_2'), ('G_1', 'Y'), ('G_1', 'P'), ('G_2', 'G_1'), ('G_2', 'G_2'), ('G_2', 'Y'), ('G_2', 'P'), ('Y', 'G_1'), ('Y', 'G_2'), ('Y', 'Y'), ('Y', 'P'), ('P', 'G_1'), ('P', 'G_2'), ('P', 'Y'), ('P', 'P') \][/tex].
In these lists:
- List 1 only lists individual marbles.
- List 2 lists combinations but not considering replacement.
- List 3 lists the correct pairs but in a nested format.
- List 4 accurately represents the sample space as described above.
Hence, List 4 correctly provides the sample space for pulling 2 marbles from the bag with replacement.
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