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Chapter 0 wrap-up

Readiness: what probability leans on

Closing thought

Every tool on this page came from an earlier course, but they now answer one shared question. How likely is an outcome you can't predict? Sets describe the outcomes you care about. Counting rules size the space they live in. Sums and series add up chances that come in a list, and integrals add up chances spread along a line.

You can now set up the raw arithmetic of a probability question without stopping to relearn it. Two dice give 6 \times 6 = 36 outcomes. Choosing 2 cards from 5 gives \binom{5}{2} = 10 hands. The series \frac12 + \frac14 + \frac18 + \cdots adds up to 1. Chapter 1 puts these tools to work. It states the three rules every probability must obey and shows that the familiar formulas follow from them.

Recipes in this chapter

Step-by-step procedures for the problem types this chapter teaches.

Evaluate a sum written in sigma notation Sigma notation and finite sums

Chapter review

One practice set per topic this chapter taught — same fixed, ungraded ladders as the practice panels inside each section.

Practice: Chapter review: sets, unions, and complements (4 problems · doesn’t affect your progress)

|A|=8, |B|=5, |A ∩ B|=3. Find |A ∪ B|.

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Why

|A ∪ B| = |A|+|B|-|A ∩ B| = 8+5-3 = 10.

Correct answer

Also accepted:

Why

10

A die is rolled. Let D = {3, 5}. What is the complement of D?

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That's D itself, not everything OUTSIDE D.

That's the whole sample space, not just what's excluded from D.

The complement is never empty unless D was the whole sample space to begin with.

Why

The complement of D is every outcome in the sample space {1,...,6} NOT in D: {1,2,4,6}.

{1, 2, 4, 6}

The universal set has 40 elements. |A|=15. Find |complement of A|.

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Why

40-15 = 25.

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Why

25

Events A and B are mutually exclusive. What must be true?

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The UNION being empty would mean neither event can ever happen at all — mutual exclusivity is about the intersection.

Their sizes have no bearing on mutual exclusivity.

Being a subset is a completely different relationship from having no shared outcomes.

Why

Mutually exclusive means A and B share no outcomes at all — exactly A ∩ B = empty set.

A ∩ B is the empty set

Practice: Chapter review: counting foundations (4 problems · doesn’t affect your progress)

A diner offers 4 entrees and 3 desserts. How many entree-dessert combinations are there?

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Why

4 × 3 = 12, by the product rule (two independent stages).

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Why

12

In how many ways can 5 distinct books be arranged in a row on a shelf?

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Why

5! = 5×4×3×2×1 = 120.

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Why

120

In how many ways can you choose an unordered group of 3 people from 8?

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Why

C(8,3) = 8!/(3!5!) = 56.

Correct answer

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Why

56

Find the binomial coefficient C(6,2).

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Why

C(6,2) = 6!/(2!4!) = 15 — the entry in row 6, position 2 of Pascal's triangle.

Correct answer

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Why

15
Practice: Chapter review: sums, series, and limits (4 problems · doesn’t affect your progress)

Evaluate sum_{i=1}^{5} i.

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Why

1+2+3+4+5 = 15.

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Why

15

Evaluate the infinite geometric series with first term a=3 and common ratio r=0.25.

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Why

For |r|<1, the sum is a/(1-r) = 3/0.75 = 4.

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Why

4

Evaluate the definite integral of 2x from 0 to 3.

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Why

The antiderivative is x². Evaluating from 0 to 3: 3²-0² = 9.

Correct answer

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Why

9

Find the limit as x -> infinity of (2x^3)/(x^3+1).

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0 would be the answer if the denominator's degree were LARGER than the numerator's — here they match.

The ratio doesn't blow up, since numerator and denominator grow at the same rate.

1/2 inverts the actual ratio of leading coefficients.

Why

Numerator and denominator both have degree 3, so the limit is the ratio of their leading coefficients: 2/1 = 2.

2

Take the chapter quiz On to Chapter 1 →
← Limits at infinity Probability as a measure →
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