To find the probability that a randomly selected moviegoer is at least 25 years old, follow these steps:
1. Determine the total number of moviegoers:
The total number of moviegoers is given as 3670.
2. Calculate the number of moviegoers aged 25 or older:
Sum the number of moviegoers in the age groups 25-44, 45-64, and 65-74:
[tex]\[
\text{Number of moviegoers aged 25 or older} = 1100 + 690 + 520 = 2310
\][/tex]
3. Set up the probability fraction:
The probability is the ratio of the number of moviegoers aged 25 or older to the total number of moviegoers:
[tex]\[
\text{Probability} = \frac{2310}{3670}
\][/tex]
4. Simplify the fraction:
To simplify [tex]\(\frac{2310}{3670}\)[/tex], find the greatest common divisor (GCD) of 2310 and 3670. Here, the GCD is 10. Divide both the numerator and the denominator by the GCD to simplify the fraction:
[tex]\[
\frac{2310 \div 10}{3670 \div 10} = \frac{231}{367}
\][/tex]
So, the probability that a randomly selected moviegoer is at least 25 years old, expressed as a simplified fraction, is:
[tex]\[
\boxed{\frac{231}{367}}
\][/tex]
