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Sagot :
Alright, let's dive into it step by step.
### Step 1: List of All Possible Ways They Can Sit
There are 4 people, and we need to find all the permutations of how they can sit in the 4 seats. The total number of permutations is calculated as [tex]\(4! = 4 \times 3 \times 2 \times 1 = 24\)[/tex].
### Step 2: Calculate Probabilities
Now let's calculate the probabilities for each event given all 24 possible permutations.
#### Probability of Dave and Natalie Sitting Together
There are 12 favorable permutations where Dave and Natalie sit together out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{12}{24} = 0.5\)[/tex].
#### Probability of Frida and Natalie Sitting Together
Similarly, there are 12 favorable permutations where Frida and Natalie sit together out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{12}{24} = 0.5\)[/tex].
#### Probability of Sitting Boy, Girl, Boy, Girl or Girl, Boy, Girl, Boy
There are 8 favorable permutations where the seating arrangement follows the boy, girl, boy, girl or girl, boy, girl, boy pattern out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{8}{24} \approx 0.3333\)[/tex].
#### Probability of Robbie Sitting Between the Girls
There are 2 favorable permutations where Robbie sits between the girls out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{2}{24} \approx 0.0833\)[/tex].
#### Probability of the Two Boys Sitting in the Middle
There are 4 favorable permutations where the two boys sit in the middle out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{4}{24} \approx 0.1667\)[/tex].
#### Probability of Natalie Sitting Between Dave and Robbie
There are 2 favorable permutations where Natalie sits between Dave and Robbie out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{2}{24} \approx 0.0833\)[/tex].
### Summary
Here’s the matching of events to their probabilities:
- The probability of Dave and Natalie sitting together: [tex]\(0.5\)[/tex]
- The probability of Frida and Natalie sitting together: [tex]\(0.5\)[/tex]
- The probability of sitting boy, girl, boy, girl or girl, boy, girl, boy: [tex]\(0.3333\)[/tex]
- The probability of Robbie sitting between the girls: [tex]\(0.0833\)[/tex]
- The probability of the two boys sitting in the middle: [tex]\(0.1667\)[/tex]
- The probability of Natalie sitting between Dave and Robbie: [tex]\(0.0833\)[/tex]
These probabilities provide the likelihood of each described event happening given the seating arrangements of Dave, Frida, Natalie, and Robbie.
### Step 1: List of All Possible Ways They Can Sit
There are 4 people, and we need to find all the permutations of how they can sit in the 4 seats. The total number of permutations is calculated as [tex]\(4! = 4 \times 3 \times 2 \times 1 = 24\)[/tex].
### Step 2: Calculate Probabilities
Now let's calculate the probabilities for each event given all 24 possible permutations.
#### Probability of Dave and Natalie Sitting Together
There are 12 favorable permutations where Dave and Natalie sit together out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{12}{24} = 0.5\)[/tex].
#### Probability of Frida and Natalie Sitting Together
Similarly, there are 12 favorable permutations where Frida and Natalie sit together out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{12}{24} = 0.5\)[/tex].
#### Probability of Sitting Boy, Girl, Boy, Girl or Girl, Boy, Girl, Boy
There are 8 favorable permutations where the seating arrangement follows the boy, girl, boy, girl or girl, boy, girl, boy pattern out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{8}{24} \approx 0.3333\)[/tex].
#### Probability of Robbie Sitting Between the Girls
There are 2 favorable permutations where Robbie sits between the girls out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{2}{24} \approx 0.0833\)[/tex].
#### Probability of the Two Boys Sitting in the Middle
There are 4 favorable permutations where the two boys sit in the middle out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{4}{24} \approx 0.1667\)[/tex].
#### Probability of Natalie Sitting Between Dave and Robbie
There are 2 favorable permutations where Natalie sits between Dave and Robbie out of the 24 total permutations.
Thus, the probability is [tex]\(\frac{2}{24} \approx 0.0833\)[/tex].
### Summary
Here’s the matching of events to their probabilities:
- The probability of Dave and Natalie sitting together: [tex]\(0.5\)[/tex]
- The probability of Frida and Natalie sitting together: [tex]\(0.5\)[/tex]
- The probability of sitting boy, girl, boy, girl or girl, boy, girl, boy: [tex]\(0.3333\)[/tex]
- The probability of Robbie sitting between the girls: [tex]\(0.0833\)[/tex]
- The probability of the two boys sitting in the middle: [tex]\(0.1667\)[/tex]
- The probability of Natalie sitting between Dave and Robbie: [tex]\(0.0833\)[/tex]
These probabilities provide the likelihood of each described event happening given the seating arrangements of Dave, Frida, Natalie, and Robbie.
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