Number of BirthsThe graph of the discrete probability distribution below represents the number of live births by a mother 50–54 years old who had a live birth in 2017.


d. If you randomly choose a 50- to 54-year-old mother who had a live birth in 2017, how many live births would you expect her to have had?

Verified step by step guidance

1. Identify the random variable X as the number of live births a 50- to 54-year-old mother had in 2017, with possible values from 1 to 10 as shown on the x-axis of the graph.
2. Note the corresponding probabilities P(X = x) for each number of live births x from the graph. These probabilities are given on the y-axis and labeled above each bar.
3. Recall that the expected value (mean) of a discrete random variable X is calculated using the formula: \[\text{E}(X) = \sum_{x} x \cdot P(X = x)\]
4. Multiply each number of live births x by its corresponding probability P(X = x) to find the weighted values.
5. Sum all these weighted values to find the expected number of live births for a randomly chosen mother aged 50 to 54 who had a live birth in 2017.

Key Concepts

Discrete Probability Distribution

A discrete probability distribution lists all possible outcomes of a random variable and their associated probabilities. Each probability is between 0 and 1, and the sum of all probabilities equals 1. This distribution helps describe the likelihood of different numbers of live births in this context.

Expected Value (Mean) of a Discrete Random Variable

The expected value is the weighted average of all possible outcomes, calculated by multiplying each outcome by its probability and summing the results. It represents the long-run average or the most likely average outcome, such as the average number of live births for mothers aged 50-54.

Interpreting Probability Graphs

Probability graphs visually display the probabilities of different outcomes, making it easier to identify the most likely events and their probabilities. Understanding how to read the graph, including the height of bars or points, is essential for calculating expected values and making informed conclusions.