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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblema 4.Q.4a
Capitolo 4, Problema 4.Q.4a

The five-year survival rate of people who undergo a liver transplant is 75%. The surgery is performed on six patients. (Source: Mayo Clinic)
a. Construct a binomial distribution.

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Step 1: Understand the problem. This is a binomial distribution problem where we are analyzing the survival rate of patients undergoing liver transplants. The probability of success (survival) is given as 75% or 0.75, and the number of trials (patients) is 6.
Step 2: Recall the formula for the binomial probability distribution. The probability of exactly k successes in n trials is given by: P(k)=(n!k!(n-k)!)×pk×(1-p)n-k, where n is the number of trials, k is the number of successes, and p is the probability of success.
Step 3: Identify the values for this problem. Here, n = 6 (number of patients), p = 0.75 (probability of survival), and k will range from 0 to 6 (number of survivors). You will calculate the probability for each value of k using the formula.
Step 4: Compute the binomial coefficients for each value of k. The binomial coefficient is calculated as (n!)/(k!×(n-k)!). For example, for k = 0, 1, 2, ..., 6, calculate the respective coefficients.
Step 5: Plug the values of k, n, and p into the binomial probability formula for each k (from 0 to 6). This will give you the probabilities for each possible number of survivors. Once calculated, organize these probabilities into a table to construct the binomial distribution.

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Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, 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).
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Mean & Standard Deviation of Binomial Distribution

Parameters of Binomial Distribution

The parameters of a binomial distribution are 'n', the number of trials, and 'p', the probability of success on each trial. For this question, 'n' is 6 (the number of patients) and 'p' is 0.75 (the survival rate). These parameters are essential for calculating probabilities related to the number of patients who survive the five-year mark.
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Mean & Standard Deviation of Binomial Distribution

Calculating Probabilities

To construct a binomial distribution, one must calculate the probabilities of different outcomes using the binomial probability formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k). Here, 'k' represents the number of successes (patients surviving), and 'n choose k' is the binomial coefficient. This allows us to determine the likelihood of various survival scenarios among the six patients.
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Probability From Given Z-Scores - TI-84 (CE) Calculator
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