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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 4.2.3
Capitolo 4, Problema 4.2.3

Graphical Analysis In Exercises 3–5, the histogram represents a binomial distribution with five trials. Match the histogram with the appropriate probability of success p. Explain your reasoning.
a. p = 0.25
b. p = 0.50
c. p = 0.75
Histogram displaying a binomial distribution with five trials, showing probabilities for values 0 to 5 on the x-axis.

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Step 1: Understand the problem. The histogram represents a binomial distribution with five trials, and we need to match it with the correct probability of success (p). The options are p = 0.25, p = 0.50, and p = 0.75.
Step 2: Recall the properties of a binomial distribution. The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p). The shape of the distribution depends on the value of p. For p = 0.25, the distribution is skewed to the right; for p = 0.50, it is symmetric; and for p = 0.75, it is skewed to the left.
Step 3: Analyze the histogram. The histogram shows that the probabilities are concentrated around higher values of x (3, 4, and 5). This indicates that the probability of success (p) is relatively high, as more successes are observed.
Step 4: Match the histogram with the correct value of p. Since the distribution is skewed to the left and the probabilities are concentrated around higher values of x, the appropriate probability of success is p = 0.75.
Step 5: Explain the reasoning. When p = 0.75, the likelihood of success in each trial is high, leading to more occurrences of higher values of x in the binomial distribution. This matches the shape of the histogram provided.

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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. It is characterized by two parameters: the number of trials (n) and the probability of success (p). The distribution is discrete, meaning it only takes on integer values from 0 to n, and is often represented graphically using histograms.
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Mean & Standard Deviation of Binomial Distribution

Probability of Success (p)

The probability of success (p) in a binomial distribution indicates the likelihood of achieving a success in a single trial. It ranges from 0 to 1, where 0 means no chance of success and 1 means certainty of success. Different values of p affect the shape of the distribution; for example, p = 0.5 typically results in a symmetric distribution, while values closer to 0 or 1 create skewed distributions.
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Introduction to Probability

Histogram Interpretation

A histogram visually represents the frequency distribution of a dataset, in this case, the probabilities of different outcomes in a binomial distribution. Each bar corresponds to the probability of achieving a specific number of successes (x) in the trials. By analyzing the heights of the bars, one can infer the most likely outcomes and how they relate to the probability of success, aiding in matching the histogram to the correct value of p.
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Intro to Histograms
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