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 10 adults who regret getting tattoos are randomly selected. a) Why can r, the number of adults who say they were too young when they got their tattoos, be considered a binomial random variable? b) What is the probability that all of the adults say they were too young? What is the most likely number of adults who say they were too young to be found? d) Which number(s) of adults who say they were too young are unlikely to be obtained in the selection?
The Correct Answer and Explanation is:
Part a) Why can r, the number of adults who say they were too young when they got their tattoos, be considered a binomial random variable?
In this case, the random variable $r$ (the number of adults who say they were too young when they got their tattoos) can be considered a binomial random variable because the situation meets all the criteria for a binomial distribution:
- Fixed number of trials: We are selecting 10 adults, so the number of trials is fixed at 10.
- Two possible outcomes: For each selected adult, there are only two possible outcomes: either they say they were too young when they got their tattoo, or they do not.
- Constant probability of success: The probability of an adult saying they were too young (success) is constant for each trial, given as 20% (0.20), as stated in the problem.
- Independence: The outcome for each adult is independent of the others, meaning the response of one adult does not influence the response of another.
Thus, $r$, the number of adults who say they were too young when they got their tattoos, follows a binomial distribution with parameters $n = 10$ (the number of trials) and $p = 0.20$ (the probability of success on each trial).
Part b) What is the probability that all of the adults say they were too young? What is the most likely number of adults who say they were too young to be found?
The probability that all 10 adults say they were too young can be calculated using the binomial probability formula:
$$
P(r = 10) = \binom{10}{10} (0.20)^{10} (1 – 0.20)^{0} = (0.20)^{10} \approx 1.024 \times 10^{-7}
$$
So, the probability that all 10 adults say they were too young is approximately $1.024 \times 10^{-7}$, which is a very small probability. This means it is extremely unlikely for all 10 adults to say they were too young.
The most likely number of adults who say they were too young can be found by determining the mode of the binomial distribution. In a binomial distribution, the most likely number of successes is often close to $np$, the expected value of the distribution. The expected number of adults who say they were too young is:
$$
\text{Expected number} = np = 10 \times 0.20 = 2
$$
Therefore, the most likely number of adults who say they were too young is 2.
Part d) Which number(s) of adults who say they were too young are unlikely to be obtained in the selection?
To determine which numbers of adults are unlikely to be obtained, we can consider the probabilities for different values of $r$. In a binomial distribution, values of $r$ that are far from the expected value (2) are less likely to occur.
- Extremely low values like 0 (no adults say they were too young) are unlikely, as this would mean that all 10 adults did not feel they were too young, which is a rare event.
- Extremely high values like 10 (all adults say they were too young) are also very unlikely, as shown in part b, where the probability of this event is extremely small.
In general, values like $r = 0$ and $r = 10$ would be considered unlikely outcomes. Most values of $r$ will be around 2 (the expected number), with decreasing probabilities as the number of adults who say they were too young moves away from 2.