Back to Exercise: Develop an empirical probability distribution

Exercises: Develop an Empirical Probability Distribution and Find Its Expected Value

Work through each section in order. To turn a frequency table into a probability distribution, estimate each probability as a RELATIVE FREQUENCY: count divided by total (or percent divided by 100). Every probability must be between 0 and 1, and they should sum to about 1. Compute the expected value as $E(X) = \sum x \cdot P(X = x)$, and scale to a population total with $n \cdot E(X)$. State any modeling choice you make for an open-ended category. Round as directed.

Grade 11·23 problems·~38 min·Common Core Math - HS Statistics and Probability·group·hss-md-a-4

Work through problems with immediate feedback

Warm-Up: Relative Frequency and Reading Tables

These problems review relative frequency and reading frequency tables.

A survey records how many pets each of 200 households owns. In the survey, 50 households own exactly 1 pet. The relative frequency of "1 pet" is which of the following?

$\frac{50}{200} = 0.25$

The count $50$

$\frac{200}{50} = 4$

The total $200$

A survey of 400 households records the number of cars per household. Exactly 120 households own 2 cars. Estimate $P(X = 2)$ as a relative frequency, written as a decimal.

A survey reports the number of TV sets per household as percentages: 0 sets 5%, 1 set 30%, 2 sets 40%, 3 sets 25%. Convert each percentage to a probability (percent divided by 100). Write $P(X = 1)$ : ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ and $P(X = 2)$ : ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ as decimals.

P(X = 1):

P(X = 2):

Fluency Practice

Build distributions, check the sum, and compute expected values.

A survey of 250 households records the number of children. The table gives raw counts.

Children	0	1	2	3
Count	50	75	100	25

Estimate $P(X = 2)$ as a relative frequency, written as a decimal.

An empirical distribution for the number of pets per household is:

Pets	0	1	2	3
Probability	0.20	0.35	0.30	0.15

Add the probabilities. What is the sum?

A reported empirical distribution lists 0 sets 2%, 1 set 35%, 2 sets 33%, 3 sets 18%, 4 or more 11%. Adding the probabilities gives $0.02 + 0.35 + 0.33 + 0.18 + 0.11 = 0.99$ . What is the best explanation?

The sum is 0.99 because each reported percentage was rounded; this is expected, not an error, and the distribution can be renormalized if an exact sum is needed.

The distribution is invalid and must be discarded.

A category is missing, so a probability was left out.

Probabilities never need to sum to 1, so 0.99 is fine and means nothing.

A distribution for the number of TVs per household has an open-ended top category "4 or more" with probability 0.12. To compute $E(X)$ , the class decides to treat "4 or more" as the representative value 4. Using that decision, what is the contribution of the "4 or more" category to $E(X)$ (that is, the value times its probability)?

A small empirical distribution is:

x	0	1	2
Probability	0.25	0.50	0.25

Compute the expected value $E(X) = \sum x \cdot P(X = x)$ .

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