At West Middle School, 30% of students have a dog, 20% of students have a cat, 10% of students have another kind of pet, and 40% of students have no pets.
a) Explain how you can use a random-number table to find the experimental probability that in a group of 5 students, at least 2 will have a dog.
b) Use your model from part a and the table below to find the experimental probability that in a group of 5 students, at least 2 will have a dog.
65926 31459 31986 65809 80462
27387 39075 46738 21986 59837
91384 10987 26491 68498 98796
32596 26448 31235 63256 53121
a) To find the experimental probability that in a group of 5 students at least 2 will have a dog using a random-number table, you can follow these steps:
1. Assign each outcome of a random number as follows:
- If the random number falls within the range 0-29, it represents a student with a dog.
- If the random number falls within the range 30-49, it represents a student with a cat.
- If the random number falls within the range 50-59, it represents a student with another kind of pet.
- If the random number falls within the range 60-99, it represents a student with no pets.
2. Generate random numbers from the table for each student. For example, let's generate 5 random numbers:
- Student 1: 65
- Student 2: 92
- Student 3: 31
- Student 4: 98
- Student 5: 75
3. Count the number of students who have a dog. In this case, there are 2 students (Student 1 and Student 5) who have a dog.
4. Repeat steps 2 and 3 multiple times, ideally thousands of times, to get a sufficient sample size for a more accurate experimental probability.
b) Now, let's use the provided random numbers to find the experimental probability that in a group of 5 students, at least 2 will have a dog.
From the given random numbers:
- Student 1: 65 (no dog)
- Student 2: 92 (no dog)
- Student 3: 31 (dog)
- Student 4: 98 (no dog)
- Student 5: 75 (dog)
In this case, there are 2 students (Student 3 and Student 5) who have a dog out of the 5 students. Hence, the experimental probability that at least 2 students will have a dog in a group of 5 students is 2/5 or 0.4.
a) To use a random-number table to find the experimental probability that in a group of 5 students, at least 2 will have a dog, you can follow these steps:
1. Assign numbers to each possibility: Write down the numbers 00 to 99, representing the two-digit numbers in the random-number table.
2. Determine the outcomes: Since the probability of having a dog is 30%, you can divide the numbers into three groups: 00-29 (representing no dog), 30-49 (representing one or more dogs), and 50-99 (representing no dog).
3. Find the desired outcomes: In this case, we want to find the probability that at least 2 out of 5 students will have a dog. That means we want to find the outcomes that represent one or more dogs (30-49) at least twice (2 or more times) in a group of 5.
4. Count the favorable outcomes: Using the random-number table, start by selecting a two-digit number and see if it falls within the range of 30-49. Repeat this process for each of the remaining four students, counting how many times you encounter a number in the range 30-49.
5. Repeat the process: To obtain a reliable experimental probability, repeat steps 4 multiple times (at least 20-30) and count the number of times you have at least 2 students with a dog out of the 5 selected.
b) Using the provided random-number table:
You can use the table to follow the steps mentioned earlier and count the number of times you encounter a number in the range 30-49 in groups of 5 students. Let's go through a few rows as an example:
Row 1: 65926
In this row, none of the numbers fall within the range 30-49, so the outcome is 0.
Row 2: 31459
In this row, two numbers (31 and 45) fall within the range 30-49, so the outcome is 1.
Row 3: 31986
In this row, two numbers (31 and 19) fall within the range 30-49, so the outcome is 1.
Row 4: 65809
In this row, none of the numbers fall within the range 30-49, so the outcome is 0.
Row 5: 80462
In this row, one number (46) falls within the range 30-49, so the outcome is 0.
You can continue this process for the remaining rows and count the total number of outcomes where there are at least 2 students with a dog out of 5 selected. Finally, divide this count by the total number of trials (rows in the table) to find the experimental probability.