Problem 17 - 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 ten households are visited at random, what is the probability that four 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: Identify the Type of Probability Distribution

This is a binomial distribution problem because we are dealing with a fixed number of trials (10 households), two possible outcomes (tuned in or not tuned in), and a constant probability of success (household tuned in).

Step 2: Define the Parameters of the Binomial Distribution

Let the random variable, X, represent the number of households tuned in to the local news. Here, the number of trials, \(n = 10\). The probability of success (tuned to the news) on each trial is \(p = 0.4\). We are interested in finding \(P(X = 4)\).

Step 3: Apply the Binomial Probability Formula

The probability of exactly \(k\) successes in \(n\) independent trials is given by: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\] Where \(\binom{n}{k}\) is the binomial coefficient calculated as \(\frac{n!}{k!(n-k)!}\).

Step 4: Calculate the Binomial Coefficient

Calculate \(\binom{10}{4}\) which is \[\frac{10!}{4! \cdot (10-4)!} = \frac{10 \cdot 9 \cdot 8 \cdot 7}{4 \cdot 3 \cdot 2 \cdot 1} = 210.\]

Step 5: Complete the Probability Calculation

Substitute the values into the formula:\[P(X = 4) = \binom{10}{4} (0.4)^4 (0.6)^{6} = 210 \times (0.4)^4 \times (0.6)^6.\]Calculate each component:- \((0.4)^4 = 0.0256\).- \((0.6)^6 = 0.046656\).Now multiply:\[210 \times 0.0256 \times 0.046656 = 0.2508\] (rounded to four decimal places).

Key Concepts

Probability of Success

In any binomial distribution experiment, the term "probability of success" refers to the likelihood that a single, specific event will result in a successful outcome during one trial. For example, in our television rating problem, the success we're interested in is a household having its TV tuned to the local evening news. This probability of success is denoted by the symbol \( p \).
The problems define \( p = 0.4 \), which means there's a 40% chance that any randomly chosen household is watching the local news. In binomial contexts, a "success" doesn't always mean something positive. Instead, it just labels the outcome we're examining.
Hence, it's critical to define what "success" means in the context of the problem first, before calculating it. Understanding the probability of success allows you to analyze the whole problem with a clearer perspective.

Binomial Probability Formula

The binomial probability formula is used to determine the likelihood of a specific number of successes out of a fixed number of independent trials. This formula is quite powerful in statistics, especially when dealing with binary outcomes.
The probability \( P(X = k) \) of getting exactly \( k \) successes in \( n \) trials is given by the formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
Here, each symbol has a special meaning:

• \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways \( k \) successes can occur out of \( n \) trials.
• \( p^k \) is the probability of the successful event happening \( k \) times.
• \((1-p)^{n-k} \) is the probability of the unsuccessful event occurring for the remaining \( n-k \) trials.

Each part of this formula builds upon another to compute a comprehensive probability that precisely matches the described scenario in the problem.

Binomial Coefficient

The binomial coefficient, an integral part of the binomial probability formula, specifies how many different ways \( k \) successes can be chosen from \( n \) trials.
Often seen in the formula as \( \binom{n}{k} \), it's calculated using:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
The expression \( n! \) ("n factorial") means multiplying all integers from 1 to \( n \). For example, when calculating the binomial coefficient for 10 trials and 4 successes:

• Calculate \( 10! = 10 \times 9 \times 8 \times 7 \dots \times 1 \)
• Calculate \( 4! = 4 \times 3 \times 2 \times 1 \)
• Calculate \( (10 - 4)! = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
• Plug into the coefficient formula:
• \( \binom{10}{4} = \frac{10!}{4!6!} = 210 \)

This value of 210 symbolizes the different ways the four tuned-in households can be chosen from ten. The binomial coefficient values are crucial for computing probabilities in binomial distributions.

Independent Trials

Independent trials in a binomial distribution are fundamental. They indicate that the outcome of one trial doesn't affect the others. Each trial stands alone, maintaining its own probability of success and failure.
This independence between trials is crucial when applying the binomial probability formula.
In the context of this problem, each household visit is an independent trial. Whether a particular household is watching the local evening news doesn't influence whether another household is watching or not.
This makes it appropriate to apply the binomial distribution formula since all trials share the same probability of success, \( p = 0.4 \). It ensures that the solutions we derive are accurate reflections of reality.
Understanding the concept of independent trials helps in making correct assumptions and choices when determining the type of probability distribution applicable to a given scenario. Always verify the independence of each event in your setup.