Problem 54 - Algebra and Trigonometry

Question

According to the National Science Foundation, in 2016 there was a \(17.2 \%\) probability that a doctoral degree awarded at a U.S. university was awarded in engineering. If a 2016 U.S. doctoral recipient is randomly selected, what is the probability that his or her degree was not in engineering?

Step-by-Step Solution

Answer

The probability is 0.828.

Step 1 - Understand the Given Probability

The problem states that there is a 17.2% probability that a doctoral degree awarded at a U.S. university is in engineering. This is written as a probability: \( P(Engineering) = 0.172 \).

Step 2 - Calculate the Complement Probability

The probability that a degree is not in engineering is the complement of the probability that it is in engineering. To find the complement, subtract the given probability from 1: \[ P(Not \, Engineering) = 1 - P(Engineering) \].

Step 3 - Perform the Calculation

Now substitute the given probability into the complement formula: \[ P(Not \, Engineering) = 1 - 0.172 \].

Step 4 - Simplify the Result

Perform the subtraction to find the final probability: \[ P(Not \, Engineering) = 0.828 \].

Key Concepts

Probability

Probability is a measure of how likely an event is to happen. It ranges between 0 (impossible event) and 1 (certain event). To express this likelihood, we use fractions, decimals, or percentages.
For example, a probability of 0.5 means there's a 50% chance of the event occurring.
In our exercise, the probability that a doctoral degree was in engineering is 17.2%, or 0.172 as a decimal.
It’s important to clearly understand what this probability represents to solve the problem correctly.

Complement Rule

The complement rule is a handy tool in probability. It helps us find the probability of an event not happening. If we know the probability of an event happening, we simply subtract it from 1 to find its complement.
Let's denote the event that a degree is in engineering as 'Engineering'. We have:

• The probability of 'Engineering' = 0.172.
• The complement event would be 'Not Engineering'.
  

According to the complement rule:

• \( P(Not \, Engineering) = 1 - P(Engineering) \)
  

Using this rule allows us to easily calculate probabilities for the opposite scenarios.

Basic Arithmetic

Basic arithmetic helps us solve everyday problems, including in probability. For our problem, we're using simple subtraction to find the complement probability.
First, recognize the probability of the doctoral degree being in engineering (0.172). Then, apply the complement rule formula:

• \( P(Not \, Engineering) = 1 - 0.172 \)

Perform the subtraction:

• 1 - 0.172 = 0.828

Thus, the probability that a randomly selected doctoral degree was not in engineering is 0.828, or 82.8%.