Problem 19 - Algebra 2

Question

Sociology A study shows that 50\(\%\) of the families in a community watch television during dinner. Suppose you select 10 families at random from this population. Find each probability. \(P(6 \text { of the } 10 \text { families watch television during dinner) }\)

Step-by-Step Solution

Answer

The probability that 6 out of the 10 randomly selected families watch television during dinner is approximately 0.205.

Step 1: Identify the Given Variables

We're told that 50\(\%\) of families watch television during dinner, so \(p = 0.50\). We're selecting 10 families, so \(n = 10\). Lastly, we want to find the probability that 6 out of the 10 families watch television during dinner, so \(x = 6\).

Step 2: Calculate Combination

Use the combination formula \(C(n, x) = n!/(x! ∗ (n - x)!)\). Substituting the values gives us \(C(10, 6) = 10!/(6! ∗ (10 - 6)!)\), which equals to 210.

Step 3: Calculate The Probability

Now substitute the known values into the binomial distribution formula. We get \(P(6) = 210 ∗ (0.5^6) ∗ (0.5^(10 - 6))\). The computed result is approximately 0.205.

Key Concepts

Combination Formula

When dealing with problems where you want to find out how many ways you can choose a certain number of objects from a larger set, you'll use the **combination formula**. This is especially relevant in probability and involves determining the number of possible selections. In our exercise, we wanted to find out how many combinations of 6 families we can choose from 10. The formula is: \[ C(n, x) = \frac{n!}{x! \, (n - x)!} \] Here, \( n \) is the total number of items to choose from, and \( x \) is the number of items you want to select. **Factorials** are used here, denoted by an exclamation mark (!), meaning you multiply a series of descending natural numbers. For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). For our specific example, we calculate \( C(10, 6) \) to discover how many different ways we can select 6 families from a pool of 10. Plugging in the numbers, we get: \[ C(10, 6) = \frac{10!}{6! \, 4!} = 210 \] This number, 210, represents all the possible ways to choose 6 families from 10, which is an essential part of solving the given probability problem.

Probability Calculation

Calculating the probability is the next crucial step in solving binomial problems. In our exercise, you want to find the probability that exactly 6 out of 10 families watch TV during dinner. Once you've determined the number of combinations (in this case, 210 from the previous step), use the **binomial probability formula**, which helps include the likelihood of each combination: \[ P(x) = C(n, x) \cdot (p^x) \cdot (1-p)^{n-x} \] Here, \( C(n, x) \) is the number of combinations, \( p \) is the probability of success on a single trial (50% or 0.5 in this case), and \( 1-p \) is the probability of failure. For our problem, substitute the known values into the formula: \[ P(6) = 210 \cdot (0.5^6) \cdot (0.5^4) \] This equation captures the essence of how likely it is to randomly choose 6 families that do watch TV with the given probability. Once calculated, the probability for this scenario turns out to be approximately 0.205.

Binomial Distribution

Binomial distribution is a foundational concept in probability and statistics. It describes the number of successes in a sequence of independent and identically distributed Bernoulli trials. For the exercise at hand, the distribution allows you to model situations where there are two possible outcomes, like success/failure or watching TV/not watching TV. Key elements of **binomial distribution**:

• **Fixed number of trials (n):** Each trial represents a single family in our example, with 10 families observed.
• **Two outcomes (watching/not watching TV):** In each trial, a family either watches TV with a probability \( p = 0.5 \), or they don't, with a probability \( 1-p = 0.5 \).
• **Constant probability of success:** The probability remains constant across trials.
• **Independence:** Each trial is independent, meaning the behavior of one family does not influence another.

By using the binomial distribution, you can determine the probability of a specific number of successes (such as 6 families) given the number of trials and the success probability per trial. In essence, it's a mathematical framework that gives structure and predictability to random processes that involve binary outcomes.