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A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In our case, we deal with a binomial distribution because each person in our survey sample has two choices: preferring Jay Leno or not.
The probability distribution is described by the number of successes (those who prefer Jay Leno) in a fixed number of trials (the sample size). For binomial distributions, we use the formula:

• \( p(x) = C(n,x) p^x (1-p)^{(n-x)} \)

Here, \( C(n, x) \) is the combination formula that determines how many ways 'x' successes can occur in 'n' trials.
In the given problem, \( n = 3 \) and \( p = 0.52 \), which are used to calculate \( p(0) \), \( p(1) \), \( p(2) \), and \( p(3) \), producing our probability distribution for the survey sample.

The mean and standard deviation are vital components in understanding any probability distribution. These statistical measures help summarize a distribution’s characteristics.

• The **mean** (\( \mu \)) of a binomial distribution tells us the expected number of successes. Calculated with the formula: \( \mu = np \).
• The **standard deviation** (\( \sigma \)) measures the spread or variability of the distribution. It is calculated using the formula: \( \sigma = \sqrt{np(1-p)} \).

In the problem, for \( n = 3 \) and \( p = 0.52 \):

• The **mean** is \( \mu = 3 \times 0.52 = 1.56 \). It indicates that, on average, 1.56 out of 3 people surveyed would prefer Jay Leno.
• The **standard deviation** is \( \sigma = \sqrt{3 \times 0.52 \times 0.48} = 0.865 \). This helps us understand how much the preferences vary from the mean.

A probability histogram is a graphical representation of the probability distribution. It makes it easy to visualize the likelihood of each possible outcome.
For our binomial distribution concerning Jay Leno's preference among three TV watchers, each outcome (0, 1, 2, 3) represents the number of people preferring Jay Leno.

• **x-axis** represents the possible outcomes (0, 1, 2, 3).
• **y-axis** shows the likelihood (probability) of each outcome.

The heights of the bars in the histogram correspond to the probabilities calculated:

• For **x = 0**, the bar height is 0.110592.
• For **x = 1**, the bar height is 0.359424.
• For **x = 2**, the bar height is 0.388992.
• For **x = 3**, the bar height is 0.140608.

These bars help in quickly visualizing the distribution, showing not just individual probabilities but also the overall tendency of the sample.