The five-year survival rate of people who undergo a liver transplant is 75%. The surgery is performed on six patients. (Source: Mayo Clinic)
b. Graph the binomial distribution using a histogram and describe its shape.

검증된 단계별 안내

Step 1: Understand the problem. This is a binomial distribution problem where the probability of success (surviving 5 years after a liver transplant) is 0.75, and the number of trials (patients) is 6. The goal is to graph the binomial distribution and describe its shape.

Step 2: Recall the formula for the binomial probability mass function (PMF): P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the number of trials, 'k' is the number of successes, 'p' is the probability of success, and (n choose k) = n! / [k! * (n-k)!]. Use this formula to calculate the probabilities for k = 0, 1, 2, ..., 6.

Step 3: Create a table of values for k (number of successes) and their corresponding probabilities P(X = k). For each value of k, substitute into the binomial PMF formula to compute the probability.

Step 4: Plot a histogram using the values of k on the x-axis (0 to 6) and the corresponding probabilities P(X = k) on the y-axis. Each bar in the histogram represents the probability of a specific number of successes.

Step 5: Analyze the shape of the histogram. Since the probability of success (0.75) is relatively high, the distribution is expected to be skewed slightly to the left, with the highest probabilities concentrated around higher values of k (e.g., 4, 5, 6).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Binomial Distribution

The binomial distribution 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 a patient surviving five years post-liver transplant, with a probability of 0.75. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. Each bin's height reflects the frequency of data points within that interval. In the context of the binomial distribution, the histogram will display the number of patients who survive versus those who do not, illustrating the probabilities of different outcomes.

Shape of the Distribution

The shape of a binomial distribution can vary based on the probability of success and the number of trials. When the probability of success is greater than 0.5, the distribution is typically skewed to the right, indicating more successes. In this scenario, with a 75% survival rate and six patients, the histogram is expected to show a right-skewed shape, reflecting a higher likelihood of more patients surviving.

The five-year survival rate of people who undergo a liver transplant is 75%. The surgery is performed on six patients. (Source: Mayo Clinic)b. Graph the binomial distribution using a histogram and describe its shape.