Lesson

Read the rule, then work each example: try the setup yourself, then tap to check one step at a time.

The rules

Sample spaces

List every possible outcome: coin + die = 2 × 6 = 12 (H1…T6); two dice = 36 pairs; three dice = 6³ = 216.

Classical probability

P(E) = n(E) ÷ n(S), reduced; 0 ≤ P ≤ 1; impossible = 0, certain = 1.

Count the outcomes in the event.
Divide by all possible outcomes.
Reduce. Empirical P = times it happened ÷ times tried.

Complement and odds

P(not E) = 1 − P(E); P(at least one) = 1 − P(none); odds against = P(not E) : P(E).

P(not E) (P(Ē) or P(Eᶜ))

"Of those who…": the group is the denominator

'Percent OF a group' / 'given' → divide by that group's total only; plain 'AND' → divide by the grand total.

Find the phrase: 'of those…', 'given that…', 'what percent of the … ' — that group is the WHOLE.
Denominator = that group's row or column total.
Numerator = the cell inside that group that matches the rest of the question.
No group named ('a person is male AND …') → denominator = grand total.
Exam Q16/17: P(read) = 1100/1520 = 72%; P(man AND no) = 251/1520 = 17%.

Traps and calculator keys are on the unit page →

Worked examples (21)

Example 1 · 2.1 #1a0/5

Experiment: Tossing a coin. Event: Receiving a head

S = ___

E = ___

Find the sample space and the specified event for the following experiment.

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Example 2 · 2.1 #1b0/5

Experiment: Rolling a die. Event: Rolling an even number

S = ___

E = ___

Find the sample space and the specified event for the following experiment.

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Example 3 · 2.1 #1c0/5

Experiment: Rolling a die. Event: Rolling a number less than 6

S = ___

E = ___

Find the sample space and the specified event for the following experiment.

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Example 4 · 2.1 #1d0/5

Experiment: Tossing a coin two times. Event: Receiving one head and one tail

S = ___

E = ___

Find the sample space and the specified event for the following experiment.

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Example 5 · 2.1 #2a0/4

an even number ___

A die is rolled. Find the probability of rolling

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Example 6 · 2.1 #2b0/4

an 8 ___

A die is rolled. Find the probability of rolling

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Example 7 · 2.1 #3a0/4

one head and one tail ___

A coin is tossed 2 times. Find the probability of getting

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Example 8 · 2.1 #3b0/4

the first toss is a head ___

A coin is tossed 2 times. Find the probability of getting

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Example 9 · 2.1 #4a0/4

a club

___

A card is drawn from an ordinary deck. Find the probability of getting

[Figure: full 52-card deck, 4 rows (CLUBS, SPADES black; HEARTS, DIAMONDS red) × 13 columns A, 2–10, J, Q, K]

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Example 10 · 2.1 #4b0/4

a 4 of diamonds

___

A card is drawn from an ordinary deck. Find the probability of getting

[Figure: full 52-card deck, 4 rows (CLUBS, SPADES black; HEARTS, DIAMONDS red) × 13 columns A, 2–10, J, Q, K]

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Example 11 · 2.1 #4c0/4

a face card (Jack, Queen, or King)

___

A card is drawn from an ordinary deck. Find the probability of getting

[Figure: full 52-card deck, 4 rows (CLUBS, SPADES black; HEARTS, DIAMONDS red) × 13 columns A, 2–10, J, Q, K]

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Example 12 · 2.1 #4c0/4

a red card

___

A card is drawn from an ordinary deck. Find the probability of getting

[Figure: full 52-card deck, 4 rows (CLUBS, SPADES black; HEARTS, DIAMONDS red) × 13 columns A, 2–10, J, Q, K]

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Example 13 · 2.1 #5a0/4

a sum of 8

___

If two dice are rolled, find the probability of getting

[Figure: all 36 outcomes (red die, yellow die) in a 6 × 6 grid]

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Example 14 · 2.1 #5b0/4

a double

___

If two dice are rolled, find the probability of getting

[Figure: all 36 outcomes (red die, yellow die) in a 6 × 6 grid]

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Example 15 · 2.1 #5c0/4

a sum less than 7

___

If two dice are rolled, find the probability of getting

[Figure: all 36 outcomes (red die, yellow die) in a 6 × 6 grid]

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Example 16 · 2.1 #5d0/4

a 3 on one die or both dice

___

If two dice are rolled, find the probability of getting

[Figure: all 36 outcomes (red die, yellow die) in a 6 × 6 grid]

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Example 17 · 2.1 #6.10/5

What is the empirical probability of rolling an 8?

___

In an experiment, two dice are rolled 50 times and the sum of the faces are recorded in a chart, as shown at the right.

Sum of Rolls of Two Dice

4 6 4 8 10 8 6 11 9 2
3 6 12 5 6 8 4 8 7 11
5 2 8 3 9 4 10 3 5 3
7 8 3 11 6 4 8 3 6 8
8 9 7 8 5 9 3 6 7 4

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Example 18 · 2.1 #6.20/4

What is the theoretical probability of rolling an 8?

___

In an experiment, two dice are rolled 50 times and the sum of the faces are recorded in a chart, as shown at the right.

Sum of Rolls of Two Dice

4 6 4 8 10 8 6 11 9 2
3 6 12 5 6 8 4 8 7 11
5 2 8 3 9 4 10 3 5 3
7 8 3 11 6 4 8 3 6 8
8 9 7 8 5 9 3 6 7 4

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Example 19 · 2.1 #80/4

A card is drawn from a deck of card. If one card is selected, find the probability that the card is NOT a four of diamonds.

___

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Example 20 · 2.1 #90/5

If one driver is randomly selected from the populations represented in the figure below, find the probability that the person is at least 20 years old.

[Figure: Pie chart 'Number of U.S. Car Drivers, by Age Group' — ≤ 19: 9 million; 20–29: 36 million; 30–39: 36 million; 40–49: 37 million; 50–59: 40 million; 60–69: 31 million; 70–79: 17 million; ≥ 80: 8 million]

___

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Example 21 · 2.1 #100/5

The winner of a raffle will receive a new car. If 100 raffle tickets were sold and you purchased 4 tickets, what are the odds against your winning the car?

___

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