11. Compound Probability, Tables, and Counting

Session 11: Compound Probability, Tables, and Counting

Learning goals

By the end of this session, you will be able to:

• Construct a sample space.
• Solve independent and dependent events.
• Use complements and counting efficiently.

Big idea

Compound probability depends on translating words such as 'and,' 'or,' and 'at least one' into a sample space or a valid probability rule.

Rich concept explanation

Compound probability involves more than one event.

For independent events, one event does not change the probability of the other. Flipping two fair coins is independent:
\[ P(H\text{ and }H)=\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}. \]

For dependent events, the first event changes the second probability. If a bag has 3 red and 2 blue marbles, the probability of drawing two red marbles without replacement is:
\[ \frac{3}{5}\cdot\frac{2}{4}=\frac{6}{20}=\frac{3}{10}. \]
The second denominator is 4 because one marble has already been removed.

For "A or B" questions, add probabilities only when the outcomes cannot happen at the same time. Rolling a 1 or 2 on one die gives \(1/6+1/6=2/6=1/3\).

The counting principle says that if one choice has \(m\) options and another has \(n\) options, the total number of ordered combinations is \(mn\).

Common misconception: multiplying for every compound problem. "And" often suggests multiplication; "or" often suggests addition, but the event structure must be checked.

Academic habit: decide whether events are independent or dependent before calculating.

A reliable data-reasoning routine

Use this five-part routine whenever a problem combines words, data, and a calculation:

1. Understand: restate the target in plain language; identify the population or event, variables, units, and any words that restrict the group.
2. Represent: organize the given information in a table, ordered list, graph, equation, sample space, or labeled probability statement.
3. Solve: choose a rule that matches the data and carry out one labeled calculation at a time without rounding too early.
4. Check: verify the denominator, units, scale, and reasonableness; then interpret the result in the original context.
5. Communicate: state what the evidence supports and avoid stronger claims, especially causal claims, that the design cannot justify.

The routine is deliberately slower than guessing at the first arithmetic operation. On the GED®, a correct setup usually saves more time than it costs because it prevents denominator, scale, and interpretation errors.

Ten real-life worked examples

The examples below move from direct application to deeper interpretation. Read the complete reasoning, cover the solution, and then solve the problem again from the prompt alone.

Worked example 1

Real-life problem. Two fair coins are flipped. Find the probability of two heads.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. List the equally likely ordered outcomes or multiply independent probabilities.

Step 3 - Solve step by step.

1. The sample space is HH, HT, TH, TT.
2. Only HH has two heads.
3. Compute 1 favorable outcome / 4 total outcomes = 1/4.
4. Equivalently, (1/2)(1/2) = 1/4.

Step 4 - Check and interpret. The four outcomes are equally likely and exactly one is favorable.

Why this matters. Multi-stage chance models appear in genetics, inspections, and repeated decisions.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 2

Real-life problem. A customer independently has a 0.8 chance of paying on time in each of two months. Find the probability of on-time payment in both months.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. For independent 'and' events, multiply the probabilities.

Step 3 - Solve step by step.

1. Let A be on-time payment in month 1 and B in month 2.
2. Independence means P(A and B) = P(A)P(B).
3. Compute 0.8 × 0.8 = 0.64.
4. The probability of on-time payment in both months is 64%.

Step 4 - Check and interpret. The probability of both events is smaller than the probability of either single event.

Why this matters. Businesses use repeated-payment probabilities when estimating cash flow.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 3

Real-life problem. A bag contains 3 red and 2 blue marbles. Two are drawn without replacement. Find the probability both are red.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Update the favorable count and total after the first draw.

Step 3 - Solve step by step.

1. The first-red probability is 3/5.
2. After one red is removed, 2 red remain among 4 marbles.
3. Multiply (3/5)(2/4) = 6/20.
4. Simplify to 3/10, or 30%.

Step 4 - Check and interpret. The second probability must be 2/4, not 3/5, because the bag changed.

Why this matters. Without-replacement models describe sampling items for inspection or selection.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 4

Real-life problem. A fair die is rolled. Find the probability of rolling a 1 or a 2.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. The outcomes cannot happen together on one roll, so add their probabilities.

Step 3 - Solve step by step.

1. P(1) = 1/6 and P(2) = 1/6.
2. The events are mutually exclusive on one roll.
3. Add 1/6 + 1/6 = 2/6.
4. Simplify to 1/3.

Step 4 - Check and interpret. Counting favorable faces directly also gives 2 out of 6.

Why this matters. Mutually exclusive categories are common in one-choice survey and routing outcomes.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 5

Real-life problem. From a standard deck, what is the probability of drawing a heart or a king? Account for overlap.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Add the two event probabilities, then subtract the king of hearts counted twice.

Step 3 - Solve step by step.

1. There are 13 hearts and 4 kings.
2. The king of hearts belongs to both events, so the simple sum double-counts one card.
3. Favorable cards = 13 + 4 - 1 = 16.
4. The probability is 16/52 = 4/13.

Step 4 - Check and interpret. Listing the union should contain the 13 hearts plus the 3 non-heart kings, totaling 16.

Why this matters. Overlap correction is used when customers or records can belong to multiple categories.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 6

Real-life problem. A uniform shop offers 4 shirts and 3 pairs of pants. How many outfits contain one shirt and one pair of pants?

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Use the fundamental counting principle for sequential choices.

Step 3 - Solve step by step.

1. There are 4 choices for the shirt.
2. For each shirt, there are 3 choices for pants.
3. Multiply 4 × 3 = 12.
4. There are 12 possible outfits.

Step 4 - Check and interpret. A 4-by-3 table would contain 12 shirt-pants cells.

Why this matters. Retailers count combinations when planning product variants and catalog displays.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 7

Real-life problem. Five finalists are available for president and vice president. How many ordered assignments are possible?

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Choose the first role, then the second role from the remaining people.

Step 3 - Solve step by step.

1. There are 5 choices for president.
2. After choosing a president, 4 people remain for vice president.
3. Multiply 5 × 4 = 20.
4. There are 20 ordered role assignments.

Step 4 - Check and interpret. Order matters because swapping the two people changes their roles.

Why this matters. Organizations use ordered counting when positions or task assignments differ.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 8

Real-life problem. A restaurant meal has 2 soup choices, 3 main-course choices, and 2 dessert choices. Find the total complete meals.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Multiply the number of options at every independent selection stage.

Step 3 - Solve step by step.

1. Start with 2 soup choices.
2. For each soup, there are 3 mains, giving 2 × 3 = 6 partial meals.
3. Each partial meal can use 2 desserts.
4. Compute 6 × 2 = 12 complete meals.

Step 4 - Check and interpret. A three-level tree would end in 12 branches.

Why this matters. Menus and product configurators rely on staged counting.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 9

Real-life problem. A weather model gives rain probability 0.3 on Saturday and 0.4 on Sunday, treated as independent. Find the probability it rains on neither day.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Use complements for each day, then multiply the independent no-rain probabilities.

Step 3 - Solve step by step.

1. P(no rain Saturday) = 1 - 0.3 = 0.7.
2. P(no rain Sunday) = 1 - 0.4 = 0.6.
3. Multiply 0.7 × 0.6 = 0.42.
4. The probability of no rain on either day is 42%.

Step 4 - Check and interpret. Both no-rain events occurring should be less likely than either one alone.

Why this matters. Weekend event planners combine multi-day weather risks.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Worked example 10

Real-life problem. A machine has a 0.1 defect probability per independent item. Find the probability that at least one of two items is defective.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Use one minus the probability of no defects.

Step 3 - Solve step by step.

1. The probability one item is not defective is 1 - 0.1 = 0.9.
2. The probability neither of two items is defective is 0.9 × 0.9 = 0.81.
3. Compute 1 - 0.81 = 0.19.
4. The probability of at least one defect is 19%.

Step 4 - Check and interpret. The result includes exactly one defect and two defects without double-counting cases.

Why this matters. Quality-control teams use complement methods to estimate batch risk.

Common trap. Multiplying or adding simply because of one keyword without checking dependence, overlap, or the complete sample space.

Session synthesis

The central idea of this session is: Compound probability depends on translating words such as 'and,' 'or,' and 'at least one' into a sample space or a valid probability rule.

Before moving on, explain one example aloud without looking at its solution. Name the target, justify the method, reproduce the calculation, check the result, and finish with a sentence in context. If any of those five parts is missing, revisit the matching step rather than memorizing only the final number.

Final accuracy checklist

• Did I answer the exact question that was asked?
• Did I use the group, total, interval, or event that belongs in the denominator?
• Did I preserve units and label percentages, percentage points, or rates correctly?
• Is the answer reasonable for the scale and context?
• Does my conclusion match the strength of the data and study design?