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QuestionA store collects age data from customers. Find these probabilities for a randomly chosen customer:
a. P(30≤age≤49)P(30 \leq \text{age} \leq 49) b. P(age<30 or age>49)P(\text{age} < 30 \text{ or age} > 49) c. P(age≥30)P(\text{age} \geq 30)

Studdy Solution

STEP 1

Assumptions1. The total number of customers is the sum of the number of customers in each age group. . Each customer has an equal chance of being chosen for the prize giveaway.

STEP 2

First, we need to find the total number of customers. We can do this by adding the number of customers in each age group.
Total Customers=∑iNumber of Customers in each age groupTotal\, Customers = \sum_{i} Number\, of\, Customers\, in\, each\, age\, group

STEP 3

Now, plug in the given values for the number of customers in each age group to calculate the total number of customers.
Total Customers=64+87+74+52+70+88Total\, Customers =64 +87 +74 +52 +70 +88

STEP 4

Calculate the total number of customers.
Total Customers=64+87+74+52+70+88=435Total\, Customers =64 +87 +74 +52 +70 +88 =435

STEP 5

To answer question a, we need to find the number of customers who are at least30 but no older than49. This is the sum of the number of customers in the age groups "30-39" and "40-49".
Number of Customers (30−49)=Number of Customers (30−39)+Number of Customers (40−49)Number\, of\, Customers\, (30-49) = Number\, of\, Customers\, (30-39) + Number\, of\, Customers\, (40-49)

STEP 6

Plug in the given values for the number of customers in the age groups "30-39" and "40-49".
Number of Customers (30−49)=74+52Number\, of\, Customers\, (30-49) =74 +52

STEP 7

Calculate the number of customers who are at least30 but no older than49.
Number of Customers (30−49)=74+52=126Number\, of\, Customers\, (30-49) =74 +52 =126

STEP 8

The probability of an event is the ratio of the number of ways the event can occur to the total number of outcomes. So, the probability that the customer is at least30 but no older than49 is the ratio of the number of customers in that age group to the total number of customers.
(30≤Age≤49)=Number of Customers (30−49)Total Customers(30 \leq Age \leq49) = \frac{Number\, of\, Customers\, (30-49)}{Total\, Customers}

STEP 9

Plug in the values for the number of customers who are at least30 but no older than49 and the total number of customers.
(30≤Age≤49)=126435(30 \leq Age \leq49) = \frac{126}{435}

STEP 10

Calculate the probability that the customer is at least30 but no older than49.
(30≤Age≤49)=126435=0.289(30 \leq Age \leq49) = \frac{126}{435} =0.289

STEP 11

To answer question b, we need to find the number of customers who are either older than49 or younger than30. This is the sum of the number of customers in the age groups "<20", "20-29", and ">=60".
Number of Customers (<30 or ≥60)=Number of Customers (<20)+Number of Customers (20−29)+Number of Customers (≥60)Number\, of\, Customers\, (<30\, or\, \geq60) = Number\, of\, Customers\, (<20) + Number\, of\, Customers\, (20-29) + Number\, of\, Customers\, (\geq60)

STEP 12

Plug in the given values for the number of customers in the age groups "<20", "20-29", and ">=60".
Number of Customers (<30 or ≥60)=64+87+88Number\, of\, Customers\, (<30\, or\, \geq60) =64 +87 +88

STEP 13

Calculate the number of customers who are either older than49 or younger than30.
Number of Customers (<30 or ≥60)=64+87+88=239Number\, of\, Customers\, (<30\, or\, \geq60) =64 +87 +88 =239

STEP 14

The probability that the customer is either older than49 or younger than30 is the ratio of the number of customers in those age groups to the total number of customers.
(Age<30 or Age≥60)=Number of Customers (<30 or ≥60)Total Customers(Age <30\, or\, Age \geq60) = \frac{Number\, of\, Customers\, (<30\, or\, \geq60)}{Total\, Customers}

STEP 15

Plug in the values for the number of customers who are either older than49 or younger than30 and the total number of customers.
(Age<30 or Age≥60)=239435(Age <30\, or\, Age \geq60) = \frac{239}{435}

STEP 16

Calculate the probability that the customer is either older than49 or younger than30.
(Age<30 or Age≥60)=239435=0.549(Age <30\, or\, Age \geq60) = \frac{239}{435} =0.549

STEP 17

To answer question c, we need to find the number of customers who are at least30. This is the sum of the number of customers in the age groups "30-39", "40-49", "50-59", and ">=60".
Number of Customers (≥30)=Number of Customers (30−39)+Number of Customers (40−49)+Number of Customers (50−59)+Number of Customers (≥60)Number\, of\, Customers\, (\geq30) = Number\, of\, Customers\, (30-39) + Number\, of\, Customers\, (40-49) + Number\, of\, Customers\, (50-59) + Number\, of\, Customers\, (\geq60)

STEP 18

Plug in the given values for the number of customers in the age groups "30-39", "40-49", "50-59", and ">=60".
Number of Customers (≥30)=74+52+70+88Number\, of\, Customers\, (\geq30) =74 +52 +70 +88

STEP 19

Calculate the number of customers who are at least30.
Number of Customers (≥30)=74+52+70+88=284Number\, of\, Customers\, (\geq30) =74 +52 +70 +88 =284

STEP 20

The probability that the customer is at least30 is the ratio of the number of customers in that age group to the total number of customers.
(Age≥30)=Number of Customers (≥30)Total Customers(Age \geq30) = \frac{Number\, of\, Customers\, (\geq30)}{Total\, Customers}

STEP 21

Plug in the values for the number of customers who are at least30 and the total number of customers.
(Age≥30)=284435(Age \geq30) = \frac{284}{435}

STEP 22

Calculate the probability that the customer is at least30.
(Age≥30)=284435=0.653(Age \geq30) = \frac{284}{435} =0.653a. The probability that the customer is at least30 but no older than49 is0.289. b. The probability that the customer is either older than49 or younger than30 is0.549. c. The probability that the customer is at least30 is0.653.

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