Question

A voter survey has determined that 15% of the residents of Barkhampsted County own an iPod. Furthermore, 85% of these iPod owners are under 30 years of age, whereas 15% are 30 or older. Only 30% of the people who don't own iPods are under 30 years of age.

What percentage of the residents who are 30 or older own iPods? Or, equivalently, what is the conditional probability that a resident known to be 30 or more years of age will own in iPod? (Give your answer as a percentage correct to two decimal places.)

_________%

Answer 1

You can start with the population being x or 100 or whatever number, as the problem given has only percentages. I have started with 10,000 as it gives me a better understanding of the numerical values. I have also done the sum in terms of x, in case that was what you were looking for.

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Let the Population of Barkhampsted County = 10,000

The number of Ipod Owners = 15% of 10,000 = 1,500

The Number of Non - I pod Owners = 10,000 - 1500 = 8,500

Ipod owners who are under 30 years of age = 85% of 1500 = 1275

Ipod owners who are 30 or older = 15% of 1500 = 225

Non Ipod Owners who are under 30 years = 30% of 8500 = 2550

Non Ipod Owners who are 30 or older = 70% of 8500 = 5950

Total Number of residents who are 30 or older = 5950 + 225 = 6175

Total number of Ipod owners who are 30 or older = 225

Therefore % of Ipod owners who are 30 or older = (225/6175) * 100 = 3.64%

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Let the Population of Barkhampsted County = x

The number of Ipod Owners = 15% of x = 0.15x

The Number of Non - I pod Owners = x - 0.15x = 0.85x

Ipod owners who are under 30 years of age = 85% of 0.15x = 0.1275x

Ipod owners who are 30 or older = 15% of 0.15x = 0.0225x

Non Ipod Owners who are under 30 years = 30% of 0.85x = 0.255x

Non Ipod Owners who are 30 or older = 70% of 0.85x = 0.595x

Total Number of residents who are 30 or older = 0.595x + 0.0225x = 0.6175x

Total number of Ipod owners who are 30 or older = 0.0225

Therefore % of Ipod owners who are 30 or older = (0.0225x / 0.6175x) * 100 = 3.64%

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