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Chapter 4: Problem 35

A sample of 400 large companies showed that 130 of them offer free health fitness centers to their employees on the company premises. If one company is selected at random from this sample, what is the probability that this company offers a free health fitness center to its employees on the company premises? What is the probability that this company does not offer a free health fitness center to its employees on the company premises? Do these two probabilities add up to \(1.0 ?\) If yes, why?

Short Answer

Expert verified
The probability that a randomly selected company offers a free health fitness center is 0.325 and the probability that it does not is 0.675. These two probabilities indeed add up to 1.0 as they represent all possible outcomes.

Step by step solution

01

Find the probability of a company offering a fitness center

To obtain this probability, divide the number of companies offering free health fitness centers by the total number of companies. In mathematical form: \[P(\text{{offers wellness center}}) = \frac{{\text{{number of companies offering wellness center}}}}{{\text{{total number of companies}}}} \]Substitute the given numbers into the formula:\[P(\text{{offers wellness center}}) = \frac{{130}}{{400}} \]
02

Simplify the fraction

To simplify the fraction, divide the numerator by the denominator:\[P(\text{{offers wellness center}}) = \frac{{130}}{{400}} = 0.325\]So, the probability that a randomly selected company offers a fitness center is 0.325.
03

Find the probability of a company not offering a fitness center

Since the only two possibilities are 'offers a fitness center' and 'does not offer a fitness center', the sum of these probabilities must be 1. So, we can find the probability of a company not offering a fitness center by subtracting the probability calculated in Step 2 from 1.\[P(\text{{does not offer wellness center}}) = 1 - P(\text{{offers wellness center}})\]Substituting the value from Step 2:\[P(\text{{does not offer wellness center}}) = 1 - 0.325 = 0.675\]
04

Check if the sum of the probabilities equals 1

Finally, let's see if the sum of the probabilities we found indeed equals 1:\[P(\text{{offers wellness center}}) + P(\text{{does not offer wellness center}}) = 0.325 + 0.675 = 1.0\]So, they do equal 1.0 as per the axiom of probability which states that the sum of probabilities of all possible outcomes must equal 1.

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