Here’s a step-by-step guide to solving the problem of finding the probability that a randomly selected customer from the phone store purchased either a standard-sized phone or a Phone II:
### Step 1: Calculate the Total Number of Customers
First, we need to find the total number of purchases:
[tex]\[
7 (\text{Mini - Phone I}) + 23 (\text{Mini - Phone II}) + 31 (\text{Mini - Phone III}) + 43 (\text{Standard - Phone I}) + 41 (\text{Standard - Phone II}) + 29 (\text{Standard - Phone III}) + 2 (\text{Maximum - Phone I}) + 17 (\text{Maximum - Phone II}) + 13 (\text{Maximum - Phone III})
\][/tex]
Summing these values, we get:
[tex]\[
206 \text{ total customers}
\][/tex]
### Step 2: Calculate the Number of Customers Who Purchased Standard-Sized Phones
We tally the number for all standard-sized phones:
[tex]\[
43 (\text{Standard - Phone I}) + 41 (\text{Standard - Phone II}) + 29 (\text{Standard - Phone III})
\][/tex]
Adding these numbers gives:
[tex]\[
113 \text{ customers}
\][/tex]
### Step 3: Calculate the Number of Customers Who Purchased Phone II
We count all purchases of Phone II:
[tex]\[
23 (\text{Mini - Phone II}) + 41 (\text{Standard - Phone II}) + 17 (\text{Maximum - Phone II})
\][/tex]
Summing these values, we get:
[tex]\[
81 \text{ customers}
\][/tex]
### Step 4: Determine the Number of Customers Who Purchased Both a Standard-Sized Phone and Phone II
We look specifically at Standard - Phone II purchases:
[tex]\[
41
\][/tex]
### Step 5: Use the Principle of Inclusion and Exclusion
To find the probability of a customer purchasing either a standard-sized phone or a Phone II, we use the formula for the union of two sets:
[tex]\[
P(\text{Standard or Phone II}) = \frac{(\text{Number of standard-sized phones}) + (\text{Number of Phone II}) - (\text{Number of both standard and Phone II})}{\text{Total number of customers}}
\][/tex]
Substituting the values:
[tex]\[
P(\text{Standard or Phone II}) = \frac{113 + 81 - 41}{206} = \frac{153}{206}
\][/tex]
### Step 6: Simplify the Fraction
We simplify the fraction [tex]\(\frac{153}{206}\)[/tex]. The simplest form of this fraction is:
[tex]\[
\frac{153}{206}
\][/tex]
Hence, the probability that a randomly selected customer purchased either a standard-sized phone or a Phone II is [tex]\(\frac{153}{206}\)[/tex].
