Problem 14 - Algebra 2 and Trigonometry

Question

In \(7-14,\) answers can be rounded to four decimal places. Statisticians investigating the Internet-surfing habits of students at a large high school discovered that only 35\(\%\) are interested in protecting their privacy online. What is the probability that none of four randomly chosen students are interested in protecting their privacy?

Step-by-Step Solution

Answer

The probability is 0.1785.

Step 1: Determine the Probability of One Student Not Being Interested

Given that 35% of students are interested in protecting their privacy, the probability that a randomly chosen student is not interested in protecting their privacy is the complement of 35%. Calculate this as 1 - 0.35, which equals 0.65.

Step 2: Calculate the Probability for Four Students

The problem asks for the probability that none of the four students are interested in protecting their privacy. This means all four students are not interested. The probability for each student not being interested is 0.65. Therefore, for all four students, the probability is \(0.65^4\).

Step 3: Compute the Probability Using Exponentiation

Now we calculate \(0.65^4\). This is computed as follows: \(0.65 \times 0.65 \times 0.65 \times 0.65\). Use a calculator to perform this calculation.

Step 4: Round the Result to Four Decimal Places

After computing \(0.65^4\), which equals approximately 0.17850625, round this value to four decimal places to get 0.1785.

Key Concepts

Complementary Probability

When dealing with probability, understanding the concept of complementary probability is quite useful. In simple terms, the complement of a probability is what remains after subtracting the probability of an event from one. If you have an event that has a certain probability (e.g., 35% chance that a student is interested in privacy), then the complementary probability is the chance that the event does not happen.
\[ P( ext{Not A}) = 1 - P( ext{A}) \]
In the example exercise, since there is a 35% chance (0.35) that a student is interested in protecting their privacy, the complementary probability is 65% (0.65) that they are not interested. Complementary probabilities are particularly useful in scenarios where you need to find the likelihood of the opposite outcome.
Whenever you encounter such problems, always remember:

• Determine the probability of the original event.
• Subtract that probability from one to get the complement.

This will give you the probability of the event not occurring.

Independent Events

Independent events in probability theory mean that the outcome of one event does not affect the outcome of another. For example, if you're interested in calculating the probability of multiple independent events occurring simultaneously (like in the given high school example), you multiply the probabilities of each individual event.
In the case of four students, each student not being interested in privacy is an independent event because one student's preference does not influence another's. The calculation looks like this: if the probability that one student is not interested is 0.65, the probability that all four are not interested is:
\[ (0.65)^4 \]
By multiplying the probability of each independent event happening, you determine the overall probability of all events occurring together. When tackling problems involving independent events, always remember:

• Ensure that events do not affect each other.
• Use multiplication to find the joint probability.

This method is efficient for calculating probabilities of several independent events occurring together.

Rounding Decimals

Rounding decimals is a valuable skill when dealing with probability calculations, especially when multiple numbers are multiplied together. Results can have many decimal places, and often, you need to simplify these for practical use or presentation.
For example, after finding the probability in an exercise scenario, which was approximately 0.17850625, rounding it is needed to answer to a specific decimal place: four decimal places in this case. This means you take the number and round it to reduce its digits:
\[ 0.17850625 \rightarrow 0.1785 \]
Here's how to round decimals:

• Look at the digit in the place immediately after the target place.
• If it's 5 or higher, round up by adding one to the target place digit.
• If it's less than 5, leave the target place digit as is.

Rounding helps in ensuring your calculations are clear and easy to interpret, especially when presenting to others or making decisions based on the results.