In Exercises 25–28, find the probabilities and answer the questions.




Too Young to Tat Based on a Harris poll, among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. Assume that five adults who regret getting tattoos are randomly selected, and find the indicated probability.


a. Find the probability that none of the selected adults say that they were too young to get tattoos.

Verified step by step guidance

1. Step 1: Identify the type of probability distribution involved. Since we are dealing with a fixed number of trials (5 adults), each with two possible outcomes (either they say they were too young or they don't), and the probability of success (saying they were too young) is constant (20% or 0.2), this is a binomial probability problem.
2. Step 2: Write the formula for the binomial probability distribution. The probability of exactly k successes in n trials is given by: $P(X\; =\; k)\; =\; C(n,\; k)\; *\; pk*\; (1\; -\; p)n\; -\; k$, where C(n, k) is the number of combinations, p is the probability of success, and (1 - p) is the probability of failure.
3. Step 3: Substitute the values into the formula. Here, n = 5 (number of adults), k = 0 (none of the adults say they were too young), and p = 0.2 (probability of saying they were too young). The formula becomes: $P(X\; =\; 0)\; =\; C(5,\; 0)\; *\; (0.2)0*\; (0.8)5$.
4. Step 4: Calculate the combination term C(5, 0). The formula for combinations is: $C(n,\; k)\; =\; n!\; /\; (k!\; *\; (n\; -\; k)!)$. For C(5, 0), this simplifies to 1 because $C(5,\; 0)\; =\; 5!\; /\; (0!\; *\; 5!)\; =\; 1$.
5. Step 5: Simplify the probability expression. Since $(0.2)0=\; 1$, the probability becomes: $P(X\; =\; 0)\; =\; 1\; *\; 1\; *\; (0.8)5$. To find the final probability, calculate $(0.8)5$.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it quantifies the chance that a randomly selected adult who regrets getting a tattoo feels they were too young when they got it. Understanding probability is essential for calculating outcomes based on given percentages, such as the 20% in this scenario.

Binomial Distribution

The binomial distribution is a statistical distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, the 'success' is defined as an adult saying they were too young to get a tattoo. This concept is crucial for solving the problem, as it allows us to calculate the probability of a specific number of successes (or failures) among the selected adults.

Complement Rule

The complement rule in probability states that the probability of an event not occurring is equal to one minus the probability of the event occurring. In this question, to find the probability that none of the selected adults say they were too young, we can use the complement of the probability that at least one adult does. This rule simplifies calculations and helps in understanding the relationship between an event and its complement.