Question

the table shows the distribution, by age, of a random sample of 3990 moviegoers ages 12-74. if one moviegoer is randomly selected from this population, find the probability, expressed as a simplified fraction, that the moviegoer is not in the 25-44 age range,
age distribution of moviegoers
ages | number
12-24 | 710
25-44 | 1150
45-64 | 790
65-74 | 1340

the probability is
(type an integer or a simplified fraction )

Explanation:

Step1: Find total number of non - 25 - 44 age moviegoers

First, we sum the number of moviegoers in the age ranges 12 - 24, 45 - 64, and 65 - 74.
The number of moviegoers in 12 - 24 is 710, in 45 - 64 is 790, and in 65 - 74 is 1340.
So the total number of non - 25 - 44 age moviegoers is \(710 + 790+1340\).
\(710+790 = 1500\), and \(1500 + 1340=2840\).

Step2: Calculate the probability

The total number of moviegoers is 3990.
The probability \(P\) that a randomly selected moviegoer is not in the 25 - 44 age range is the number of non - 25 - 44 age moviegoers divided by the total number of moviegoers.
So \(P=\frac{2840}{3990}\).
We simplify this fraction by dividing both the numerator and the denominator by 10: \(\frac{284}{399}\). We check if 284 and 399 have a common factor. The factors of 284 are \(2\times2\times71\), and the factors of 399 are \(3\times133 = 3\times7\times19\). There are no common factors other than 1, so the simplified fraction is \(\frac{284}{399}\)? Wait, wait, let's recalculate the sum of non - 25 - 44. Wait, 710+790 = 1500, 1500 + 1340=2840? Wait, 710+790 is 1500, 1500+1340: 1500+1300 = 2800, 2800 + 40=2840. Total is 3990. Wait, alternatively, we can calculate the number of moviegoers in 25 - 44 is 1150, so the number of non - 25 - 44 is \(3990 - 1150=2840\). Then \(\frac{2840}{3990}=\frac{284}{399}\)? Wait, no, 2840÷10 = 284, 3990÷10 = 399. Wait, but 399 - 284 = 115, and 1150 is the number in 25 - 44. Wait, maybe I made a mistake. Wait, 710+790+1340: 710+790 = 1500, 1500+1340 = 2840. 3990-1150 = 2840. Then 2840/3990. Let's divide numerator and denominator by 10: 284/399. Wait, but 284 and 399: 399 = 3×133 = 3×7×19, 284 = 4×71. No common factors. Wait, but wait, 710+790+1340: 710+790 is 1500, 1500+1340 is 2840. 3990 is the total. So 2840/3990. Let's check 2840÷10 = 284, 3990÷10 = 399. Alternatively, maybe I miscalculated the sum. Wait, 12 - 24:710, 25 - 44:1150, 45 - 64:790, 65 - 74:1340. Sum all: 710+1150 = 1860, 1860+790 = 2650, 2650+1340 = 3990. Correct. So non - 25 - 44 is 3990 - 1150 = 2840. So probability is 2840/3990. Simplify: divide numerator and denominator by 10: 284/399. Wait, but 284 and 399: GCD(284,399). Let's use Euclidean algorithm. GCD(399,284). 399 = 1×284+115. GCD(284,115). 284 = 2×115 + 54. GCD(115,54). 115 = 2×54+7. GCD(54,7). 54 = 7×7+5. GCD(7,5). 7 = 1×5+2. GCD(5,2). 5 = 2×2+1. GCD(2,1)=1. So GCD is 1. So the simplified fraction is 284/399? Wait, no, wait 2840/3990: 2840 = 284×10, 3990 = 399×10. So yes, 284/399. Wait, but maybe I made a mistake in the sum of non - 25 - 44. Wait, 12 - 24 is 710, 45 - 64 is 790, 65 - 74 is 1340. 710+790 = 1500, 1500+1340 = 2840. Correct. So the probability is 2840/3990 = 284/399. Wait, but 2840 divided by 10 is 284, 3990 divided by 10 is 399. Alternatively, maybe 2840 and 3990 can be divided by 10, and then check if 284 and 399 have a common factor. As we saw, GCD is 1. So the simplified fraction is 284/399? Wait, no, wait 399 - 284 = 115, and 1150 is the number in 25 - 44. Wait, 1150 is 10×115, 2840 is 10×284, 3990 is 10×399. So 284/399 is correct.

Answer:

\(\frac{284}{399}\)