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Sagot :
Sure! Let's go through the probability calculations step by step:
### Given:
- There are 5 different Geography books.
- There are 6 different History books.
- The total number of books is [tex]\(5 + 6 = 11\)[/tex].
### (1) Probability that the first and last positions will be occupied by Geography books
To find this probability, we need to consider the favorable positions for the Geography books and the total possible positions.
- Total possible books in the first position: 11 (any of the 11 books can be placed).
- Remaining books for the last position: 10 (after placing the first book, there are 10 books left).
For favorable cases:
- Geography books in the first position: 5 (any of the 5 Geography books can be placed first).
- Geography books for the last position: 4 (after placing one Geography book first, there are 4 remaining).
Thus, the number of favorable outcomes is [tex]\( 5 \times 4 = 20 \)[/tex].
Total possible outcomes for the arrangement is [tex]\( 11 \times 10 = 110 \)[/tex].
Therefore, the probability is:
[tex]\[ \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{20}{110} = \frac{2}{11} \approx 0.1818 \][/tex]
### (2) Probability that all the Geography books will be grouped together
We can treat the 5 Geography books as a single entity or "super book." Thus, we now have:
- 1 group of Geography books + 6 individual History books = 7 "books" or entities to arrange.
The number of ways to arrange these 7 entities is:
[tex]\[ 7! \][/tex]
Within the Geography group, the 5 Geography books can be arranged among themselves in:
[tex]\[ 5! \][/tex]
Therefore, the number of favorable outcomes is:
[tex]\[ 7! \times 5! \][/tex]
The total number of ways to arrange all 11 books is:
[tex]\[ 11! \][/tex]
The probability is then:
[tex]\[ \frac{7! \times 5!}{11!} \approx 0.01515 \][/tex]
### (3) Probability that all the Geography books will be grouped together and all the History books will be grouped together
If all Geography books are grouped together and all History books are grouped together, we can treat each group as a single entity.
Thus, we have 2 entities (one big G group and one big H group):
- These 2 entities can be arranged in [tex]\(2!\)[/tex] ways.
Within the Geography group, the 5 Geography books can be arranged among themselves in:
[tex]\[ 5! \][/tex]
Within the History group, the 6 History books can be arranged among themselves in:
[tex]\[ 6! \][/tex]
The number of favorable outcomes is:
[tex]\[ 2! \times 5! \times 6! \][/tex]
The total number of ways to arrange all 11 books is:
[tex]\[ 11! \][/tex]
The probability is then:
[tex]\[ \frac{2! \times 5! \times 6!}{11!} \approx 0.0043 \][/tex]
### (4) Probability that all the Geography books will be first, followed by all the History books
There is exactly one way to arrange all Geography books first and then all History books last.
Within the Geography section, the 5 Geography books can be arranged in:
[tex]\[ 5! \][/tex]
Within the History section, the 6 History books can be arranged in:
[tex]\[ 6! \][/tex]
The number of favorable outcomes is:
[tex]\[ 5! \times 6! \][/tex]
The total number of ways to arrange all 11 books is:
[tex]\[ 11! \][/tex]
The probability is then:
[tex]\[ \frac{5! \times 6!}{11!} \approx 0.00216 \][/tex]
### Summary:
1. The probability that the first and last positions will be occupied by Geography books is approximately [tex]\(0.1818\)[/tex].
2. The probability that all the Geography books will be grouped together is approximately [tex]\(0.01515\)[/tex].
3. The probability that all the Geography books will be grouped together and all the History books will be grouped together is approximately [tex]\(0.0043\)[/tex].
4. The probability that all the Geography books will be first, followed by all the History books is approximately [tex]\(0.00216\)[/tex].
### Given:
- There are 5 different Geography books.
- There are 6 different History books.
- The total number of books is [tex]\(5 + 6 = 11\)[/tex].
### (1) Probability that the first and last positions will be occupied by Geography books
To find this probability, we need to consider the favorable positions for the Geography books and the total possible positions.
- Total possible books in the first position: 11 (any of the 11 books can be placed).
- Remaining books for the last position: 10 (after placing the first book, there are 10 books left).
For favorable cases:
- Geography books in the first position: 5 (any of the 5 Geography books can be placed first).
- Geography books for the last position: 4 (after placing one Geography book first, there are 4 remaining).
Thus, the number of favorable outcomes is [tex]\( 5 \times 4 = 20 \)[/tex].
Total possible outcomes for the arrangement is [tex]\( 11 \times 10 = 110 \)[/tex].
Therefore, the probability is:
[tex]\[ \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{20}{110} = \frac{2}{11} \approx 0.1818 \][/tex]
### (2) Probability that all the Geography books will be grouped together
We can treat the 5 Geography books as a single entity or "super book." Thus, we now have:
- 1 group of Geography books + 6 individual History books = 7 "books" or entities to arrange.
The number of ways to arrange these 7 entities is:
[tex]\[ 7! \][/tex]
Within the Geography group, the 5 Geography books can be arranged among themselves in:
[tex]\[ 5! \][/tex]
Therefore, the number of favorable outcomes is:
[tex]\[ 7! \times 5! \][/tex]
The total number of ways to arrange all 11 books is:
[tex]\[ 11! \][/tex]
The probability is then:
[tex]\[ \frac{7! \times 5!}{11!} \approx 0.01515 \][/tex]
### (3) Probability that all the Geography books will be grouped together and all the History books will be grouped together
If all Geography books are grouped together and all History books are grouped together, we can treat each group as a single entity.
Thus, we have 2 entities (one big G group and one big H group):
- These 2 entities can be arranged in [tex]\(2!\)[/tex] ways.
Within the Geography group, the 5 Geography books can be arranged among themselves in:
[tex]\[ 5! \][/tex]
Within the History group, the 6 History books can be arranged among themselves in:
[tex]\[ 6! \][/tex]
The number of favorable outcomes is:
[tex]\[ 2! \times 5! \times 6! \][/tex]
The total number of ways to arrange all 11 books is:
[tex]\[ 11! \][/tex]
The probability is then:
[tex]\[ \frac{2! \times 5! \times 6!}{11!} \approx 0.0043 \][/tex]
### (4) Probability that all the Geography books will be first, followed by all the History books
There is exactly one way to arrange all Geography books first and then all History books last.
Within the Geography section, the 5 Geography books can be arranged in:
[tex]\[ 5! \][/tex]
Within the History section, the 6 History books can be arranged in:
[tex]\[ 6! \][/tex]
The number of favorable outcomes is:
[tex]\[ 5! \times 6! \][/tex]
The total number of ways to arrange all 11 books is:
[tex]\[ 11! \][/tex]
The probability is then:
[tex]\[ \frac{5! \times 6!}{11!} \approx 0.00216 \][/tex]
### Summary:
1. The probability that the first and last positions will be occupied by Geography books is approximately [tex]\(0.1818\)[/tex].
2. The probability that all the Geography books will be grouped together is approximately [tex]\(0.01515\)[/tex].
3. The probability that all the Geography books will be grouped together and all the History books will be grouped together is approximately [tex]\(0.0043\)[/tex].
4. The probability that all the Geography books will be first, followed by all the History books is approximately [tex]\(0.00216\)[/tex].
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