Problem 15 - Algebra and Trigonometry

Question

Television Ratings According to a ratings survey, 40\(\%\) of the households in a certain city tune in to the local evening TV news. If 10 households are visited at random, what is the probability that 4 of them will have their television tuned to the local news?

Step-by-Step Solution

Answer

The probability is approximately 0.2508, or 25.08%.

Step 1: Understand the Problem

We need to find the probability that exactly 4 out of 10 randomly chosen households are watching the local evening news. We know that 40% of households watch the local news.

Step 2: Identify the Distribution

Since we are dealing with a fixed number of trials (10 households), and each household has two possible outcomes (watching or not watching the news), we can use the binomial distribution to solve this problem.

Step 3: Define Variables for Binomial Distribution

Let the random variable \( X \) represent the number of households that watch the local news. For a binomial distribution, we define \( n = 10 \) (number of trials), \( p = 0.40 \) (probability of success, i.e., a household has their TV on the local news), and \( k = 4 \) (number of successes we are interested in).

Step 4: Use the Binomial Probability Formula

The probability of getting exactly \( k \) successes (households watching the news) out of \( n \) trials is given by the formula: \[ P(X = k) = \binom{n}{k} p^k(1-p)^{n-k} \]Substitute the given values into the formula:\[ P(X = 4) = \binom{10}{4} (0.40)^4 (0.60)^6 \]

Step 5: Calculate the Binomial Coefficient

The binomial coefficient \( \binom{n}{k} \) is calculated as:\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = 210 \]

Step 6: Calculate the Probability

Substitute the binomial coefficient and probabilities into the formula:\[ P(X = 4) = 210 \times (0.40)^4 \times (0.60)^6 \] Calculate each part:\[(0.40)^4 = 0.0256, \quad (0.60)^6 = 0.046656 \] Now, perform the multiplication:\[ P(X = 4) = 210 \times 0.0256 \times 0.046656 = 0.2508 \]

Key Concepts

Probability

Probability is the measure of the likelihood that an event will occur. It provides a way to quantify uncertainty. In any given scenario, the probability is a number between 0 and 1, where 0 indicates impossibility and 1 signifies certainty.

For instance, suppose we randomly pick a household and want to know if they watch the evening news. If 40% of all households watch it, the probability that a randomly picked household tunes in is 0.40.

When multiple events occur, such as selecting 10 households, we can use probability to determine the chance of a specific outcome, like exactly 4 out of 10 houses watching the news.

This is where the concept of probability shines—it helps make informed predictions about everyday scenarios by using known statistical data.

Binomial Coefficient

The binomial coefficient is a key part of solving problems with the binomial distribution. It answers the question: "In how many ways can \(k\) successes occur in \(n\) trials?"

The binomial coefficient is represented as \( \binom{n}{k} \), which mathematically is calculated with the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where \(!\) denotes factorial, representing the product of all positive integers up to a number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

In our specific case, with 10 households, figuring out the formula for \( \binom{10}{4} \) helps us see the number of combinations where exactly 4 houses are tuned in to the news.

This step is crucial because it lays the foundation for applying probabilities to different scenarios, by quantifying the number of ways an outcome can happen.

Binomial Probability Formula

The binomial probability formula is the core tool for calculating the probability of a certain number of successes in a series of independent trials.

In the formula:
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

• \(P(X = k)\) represents the probability of getting exactly \(k\) successes (in our case, households watching the news).
• \(\binom{n}{k}\) is the binomial coefficient, showing how many ways \(k\) successes can occur in \(n\) trials.
• \(p\) is the probability of success on a single trial (one household tuning in), here 0.40.
• \((1-p)\) is the probability of failure (household not watching), which is 0.60.
• \(n\) is the total number of trials.\( k\) is the number of successes we're interested in.

By introducing these values into the formula, we compute the likelihood of having exactly 4 out of 10 households watching the news.

This powerful formula allows us to predict probabilities for various scenarios like this by systematically incorporating all crucial factors involved.