The probability distribution of students categorized by age...

The probability distribution of students categorized by age that visit a certain movie house on weekends is given on the right. The probabilities for $18-$ and 19 -year-olds are missing. We know that $$P(x=18)=2 P(x=19)$$. (GRAPHS CANNOT COPY) a) Complete the histogram and describe the distribution. b) Find the expected value and the variance.

The probability distribution of students categorized by age that visit a certain movie house on weekends is given on the right. The probabilities for $18-$ and 19 -year-olds are missing. We know that
$$P(x=18)=2 P(x=19)$$.
(GRAPHS CANNOT COPY)
a) Complete the histogram and describe the distribution.
b) Find the expected value and the variance.

Key Concepts

Normalization of Probability Distributions

In any probability distribution, the sum of all the probabilities of all possible outcomes must equal 1. This normalization condition is used to solve for unknown probabilities when additional constraints, such as relationships between probabilities, are provided.

Probability Distribution

A probability distribution assigns a probability to each possible outcome of a random variable. In the context of this problem, each age category of students represents an outcome, and the associated probability indicates the likelihood of that age group visiting the movie house. The distribution can be represented visually using histograms.

Ratio and Proportion of Probabilities

When relationships between the probabilities of different outcomes are given, such as one probability being a multiple of another, these ratios are used along with the normalization condition to determine the unknown probabilities. Establishing and solving equations based on these ratios is a common technique in probability problems.

Histogram Representation

A histogram is a graphical representation of a probability distribution for discrete data. Each bar in the histogram corresponds to an outcome (in this case, an age group) and its height represents the probability of that outcome. Completing a histogram involves correctly identifying the probabilities for all categories and ensuring the visual representation matches the numeric distribution.

Expected Value

The expected value of a random variable in a probability distribution is the long-run average or mean of the outcomes weighted by their probabilities. It is calculated as the sum of the product of each outcome and its corresponding probability. This measure provides a central tendency for the distribution.

Variance

Variance measures the spread or dispersion of a probability distribution around its mean. It is computed as the expected value of the squared deviations from the mean. Variance gives insight into the variability of the distribution and is useful for understanding the extent to which outcomes deviate from the expected value over repeated trials.

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