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← Unit 7 · Multiplication rules & conditional probability

Lesson

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

The rules

AND, independent events

Independent (one doesn't change the other: coin & die, with replacement, a fixed %) → P(A and B) = P(A) × P(B).

  1. Coin + die: tail AND even = 1/2 × 3/6 = 1/4 (exam Q15).
  2. Repeated % events: none of 4 text (35% do) = (.65)⁴ = .179.
  3. Both drives fail at 3% each: .03² = .0009.

AND, without replacement

Without replacement the pool shrinks: P(A and B) = P(A) × P(B given that A has occurred).

P(A and B) = P(A) · P(B given that A has occurred) (P(A ∩ B) = P(A) · P(B | A))

"AND" vs "GIVEN" in a two-way table

AND → one cell ÷ the grand total. GIVEN A → that cell ÷ A's row or column total only.

P(B given that A has occurred) = P(A and B) ÷ P(A) (P(B | A) = P(A ∩ B) / P(A))

  1. Underline the word after 'given that' (or 'of those who'). That group is your new whole.
  2. Denominator = that row/column TOTAL.
  3. Numerator = the cell in that row/column that matches the other event.
  4. No 'given'? AND → cell ÷ grand total; OR → (row + column − cell) ÷ grand total.
  5. From percents: 25% passed both, 42% passed the first → P(2nd given 1st) = .25 ÷ .42 = 60%.
Traps and calculator keys are on the unit page →

Worked examples (14)

Example 1 · 2.3 #1a0/4

Event A: Riding a horse. Event B: Getting a promotion

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Determine if the events are independent.

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Example 2 · 2.3 #1b0/4

Event A: Being rich. Event B: Driving a BMW car

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Determine if the events are independent.

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Example 3 · 2.3 #2a0/4

getting a red ball with replacement and a blue ball.

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An urn contains 5 balls (2 red, 1 blue, 1 green, and 1 white). Find the probability of

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Example 4 · 2.3 #2b0/4

getting a red ball without replacement and blue ball.

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An urn contains 5 balls (2 red, 1 blue, 1 green, and 1 white). Find the probability of

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Example 5 · 2.3 #3a0/4

Getting 3 queens

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Three cards are drawn from an ordinary deck and not replaced. Find the probability of these events

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Example 6 · 2.3 #3b0/4

Getting 3 clubs

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Three cards are drawn from an ordinary deck and not replaced. Find the probability of these events

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Example 7 · 2.3 #3c0/4

Getting a club, a spade, and a heart in order.

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Three cards are drawn from an ordinary deck and not replaced. Find the probability of these events

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Example 8 · 2.3 #4a0/3

If you store all of your computer data on a single hard drive disk, what is the probability that the drive will fail during a year?

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Assume that there is a 3% rate of disk drive failure in a year.

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Example 9 · 2.3 #4b0/4

If copies of all of your computer data are stored on two different hard disk drives, what is the probability that both drives will fail during a year?

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Assume that there is a 3% rate of disk drive failure in a year.

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Example 10 · 2.3 #5a0/4

Find the probability that a person is married given that the person is male.

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In 2015, there were approximately 254 million Americans ages 15 or older. The table shows the distribution, by marital status and gender, of this population. Numbers in the table are expressed in millions.

Marital Status of the U.S. Population, Ages 15 or Older, 2015, in Millions

MarriedNever MarriedDivorcedWidowedTotal
Male6643113123
Female67381511131
Total133812614254

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Example 11 · 2.3 #5b0/4

Find the probability that a person is female given that the person is widowed.

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In 2015, there were approximately 254 million Americans ages 15 or older. The table shows the distribution, by marital status and gender, of this population. Numbers in the table are expressed in millions.

Marital Status of the U.S. Population, Ages 15 or Older, 2015, in Millions

MarriedNever MarriedDivorcedWidowedTotal
Male6643113123
Female67381511131
Total133812614254

Try the setup yourself first, then tap.

Example 12 · 2.3 #6a0/4

Find the probability of selecting a shape with an odd number on it.

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Suppose the numbered shapes are placed in a box, and one is selected at random.

[Figure: six numbered shapes — triangles labeled 1, 3, 6; five-pointed stars labeled 2, 4, 5]

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Example 13 · 2.3 #6b0/4

Find the probability of selecting a shape with an odd number on it, given that a star was selected.

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Suppose the numbered shapes are placed in a box, and one is selected at random.

[Figure: six numbered shapes — triangles labeled 1, 3, 6; five-pointed stars labeled 2, 4, 5]

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Example 14 · 2.3 #6c0/4

Find the probability of selecting a triangle given that the shape selected has an even number on it

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Suppose the numbered shapes are placed in a box, and one is selected at random.

[Figure: six numbered shapes — triangles labeled 1, 3, 6; five-pointed stars labeled 2, 4, 5]

Try the setup yourself first, then tap.

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