Probability Models and Inference: First Steps

A probability model describes possible outcomes and their probabilities. Inference uses sample information to reason about a population while acknowledging uncertainty and the conditions of the method.

2 worked examples and 3 practice questions, with explained answers. No sign-in required.

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A reliable method

State the random variable or population parameter.
Check model conditions rather than choosing a formula by appearance.
Compute using exact probabilities or the requested approximation.
Interpret the answer in context and distinguish an estimate from certainty.

Example 1

A binomial model has 20 independent trials and success probability 0.3 per trial. Find the expected number of successes.

For a binomial model, expected count is n times p.
Use n=20 and p=0.3.
20 times 0.3 is 6; an expectation need not be the actual observed count.

Answer: \(6\)

Example 2

A binomial model has n=10 and p=0.2. Find its variance.

For independent equal-probability trials use variance n p (1-p).
The failure probability is 1-0.2=0.8.
10 times 0.2 times 0.8 is 1.6. Standard deviation would be the square root of this variance.

Answer: \(8/5\)

Common mistakes

A confidence interval is not the range containing every individual observation.
A small p-value is not the probability that the null hypothesis is true.

Practice without the answer first

A population mean is estimated as 40 with a reported margin of error 2.5. Give the interval.
A bag has 4 red and 6 blue counters. Find the probability that two successive random draws are both red, without replacement.
A bag has 3 red and 7 blue counters. Find the probability that two successive random draws are both red, replacing the first counter before drawing again.

Answers and step-by-step explanations

Practice 1

Keep units consistent.
Subtract and add 2.5 to 40.
The interval is 37.5 to 42.5; check the reported confidence level and sampling conditions separately.

Answer: \([37.5,42.5]\)

Practice 2

First red probability: \(\frac{2}{5}\).
For the second draw there is one fewer red and one fewer total counter.
Multiply conditional probabilities: \(\frac{2}{15}\).

Answer: \(\frac{2}{15}\)

Practice 3

First red probability: \(\frac{3}{10}\).
For the second draw the bag has its original counts because of replacement.
Multiply conditional probabilities: \(\frac{9}{100}\).

Answer: \(\frac{9}{100}\)

These original skill exercises are not official exam questions or a score prediction. Use the coverage notes on your exam hub to choose relevant practice.