Answer :
Let's solve the problems step-by-step:
### Part a: Probability that the customer is at least 30 but no older than 49
To find the probability, we first need to determine the number of customers in the age range 30-39 and 40-49. The table provides this information:
- Customers aged 30-39: 60
- Customers aged 40-49: 55
Combining these groups, the number of customers aged 30 to 49 is:
[tex]\[ 60 + 55 = 115 \][/tex]
To find the probability, we divide this number by the total number of customers, which is 414.
Thus, the probability is:
[tex]\[ \frac{115}{414} \approx 0.2778 \][/tex]
### Part b: Probability that the customer is either younger than 40 or older than 59
First, let's determine the number of customers in two specific groups:
- Customers younger than 40 (includes \(<20\), \(20-29\), and \(30-39\)):
- \(<20\): 97
- \(20-29\): 65
- \(30-39\): 60
Adding these values together:
[tex]\[ 97 + 65 + 60 = 222 \][/tex]
- Customers aged 60 and older:
- \( \geq 60 \): 68
Combining these groups (younger than 40 or 60 and older), the number of customers is:
[tex]\[ 222 + 68 = 290 \][/tex]
To find the probability, we divide this number by the total number of customers, which is 414.
Thus, the probability is:
[tex]\[ \frac{290}{414} \approx 0.7005 \][/tex]
### Part c: Probability that the customer is at least 60
The number of customers aged 60 and older is directly given as 68.
To find the probability, we divide this number by the total number of customers, which is 414.
Thus, the probability is:
[tex]\[ \frac{68}{414} \approx 0.1643 \][/tex]
### Final Probabilities:
- a. The probability that the customer is at least 30 but no older than 49 is \( \approx 0.2778 \)
- b. The probability that the customer is either older than 59 or younger than 40 is \( \approx 0.7005 \)
- c. The probability that the customer is at least 60 is [tex]\( \approx 0.1643 \)[/tex]
### Part a: Probability that the customer is at least 30 but no older than 49
To find the probability, we first need to determine the number of customers in the age range 30-39 and 40-49. The table provides this information:
- Customers aged 30-39: 60
- Customers aged 40-49: 55
Combining these groups, the number of customers aged 30 to 49 is:
[tex]\[ 60 + 55 = 115 \][/tex]
To find the probability, we divide this number by the total number of customers, which is 414.
Thus, the probability is:
[tex]\[ \frac{115}{414} \approx 0.2778 \][/tex]
### Part b: Probability that the customer is either younger than 40 or older than 59
First, let's determine the number of customers in two specific groups:
- Customers younger than 40 (includes \(<20\), \(20-29\), and \(30-39\)):
- \(<20\): 97
- \(20-29\): 65
- \(30-39\): 60
Adding these values together:
[tex]\[ 97 + 65 + 60 = 222 \][/tex]
- Customers aged 60 and older:
- \( \geq 60 \): 68
Combining these groups (younger than 40 or 60 and older), the number of customers is:
[tex]\[ 222 + 68 = 290 \][/tex]
To find the probability, we divide this number by the total number of customers, which is 414.
Thus, the probability is:
[tex]\[ \frac{290}{414} \approx 0.7005 \][/tex]
### Part c: Probability that the customer is at least 60
The number of customers aged 60 and older is directly given as 68.
To find the probability, we divide this number by the total number of customers, which is 414.
Thus, the probability is:
[tex]\[ \frac{68}{414} \approx 0.1643 \][/tex]
### Final Probabilities:
- a. The probability that the customer is at least 30 but no older than 49 is \( \approx 0.2778 \)
- b. The probability that the customer is either older than 59 or younger than 40 is \( \approx 0.7005 \)
- c. The probability that the customer is at least 60 is [tex]\( \approx 0.1643 \)[/tex]
