34: The Multiplication Rule

1
3-4 The Multiplication Rule

• P(A ? B) ? P(A and B) ? the probability that
  event A occurs on the first trial, AND, event B
  occurs on the second trial.
• Note We are conducting 2 trials performing 2
  tasks.

2
Exam

• True or False The probability of being struck
  by lightning is greater than the probability of
  winning a state lottery.
• The Pearson Product Moment is named after
• Karl Marx
• Carl Gauss
• Karl Pearson
• Carly Simon
• Mario Triola

3
Sample space for Exam

• S 10

4
Intuitively

• Since each question has only 1 correct answer,
  the probability of guessing correctly on both
  questions is
• 1/10

5
Theoretically

• Let A the probability of guessing correctly on
  the 1st question
• P(A) 1/2
• Let B the probability of guessing correctly on
  the 2nd question
• P(B) 1/5
• P(A and B) P(A)?P(B)
• (1/2) ?(1/5) 1/10

6
Definition

• Two events A and B are independent if the
  occurrence of one does not affect the probability
  of the occurrence of the other. (May include
  more than 2 events).
• If A and B are not independent, then they are
  said to be dependent events.

7
Explanation (?)

• The example of guessing on a test the 2
  questions are independent
• The outcome (or answer) of the first question has
  no bearing on the probabilty of guessing
  correctly on the second.
• Drawing 2 names out of a hat is dependent ..
• Drawing the first name reduces the number of
  names in the hat, thus decreasing the probability

8
Traffic Signal Lenses

• 2 lenses are randomly selected from a box of 12
  5 red 3 green 4 yellow. Find the probability
  that A the first selection is a red lens and B
  the 2nd selection is a green lens.
• P(A) 5/12
• P(B) 3/12 1/4

9
But

• We must assume that I did select a red lens on
  the first selection
• This leaves only 11 lenses in the box
• P(B) 3/11

10
Formal Multiplication Rule

• P(BA) ? the probabilaty of B given A
• i.e., the probability of B occuring given that A
  occured

11
2 New Concepts

• With replacement
• Without replacement

12
Example

• I draw 2 cards from a standard deck of 52 .
• With replacement
• I would draw the first card replace re-shuffle
  and draw the 2nd card.
• Without replacement
• I draw the first card (leaving 51 cards in the
  deck) then draw the second card.

13
Draw 2

• What is the probability of drawing A a 7 of
  spades and B heart?
• With Replacement
• P(A) 1/52
• P(B) 13/52 1/4
• P(A and B) (1/52) ? (1/4) 1/208
• Or approx 0.00481

14
Draw 2

• What is the probability of drawing A a 7 of
  spades and B heart?
• Without replacement
• P(A) 1/52
• P(BA) 13/51
• (1/52) ? (13/51) (1/4) ? (1/51) 1/204
• Approx 0.00490

15
A draw ace of spacesB draw ace of hearts.

• Find P(A ? B)
• With replacement
• Without replacement

16
Draw 4

• P(4 kings)
• P(king and king and king and king)
• With replacement
• Without replacement

17
Flip a coin Roll a die.

• P(roll 5 and get heads)
• P(roll even number and get tails)

18
Flip a coin 5 times

• P(5 tails)
• P(tail and tail and tail and tail and tail)

19
An 80 free throw shooter

• Goes to the free-throw line on a one and one.
• P(make both shots)?
• P(make shot and make shot)

20
Jury Pool

• A pool of potential jurors consists of 10 men and
  15 women.
• What is the probability that the first 2
  selections are men.
• With replacement
• Without replacement

21
Jury Pool

• What is the probability that the first selection
  is a male and the second selection is a female?
• With replacement
• Without replacement

22
Quality Control

• A QC manager claims that a in new process for
  manufacturing cameras, the rate of defects is 5.
  Out of a run of 1,000 cameras, 12 are tested and
  no defects are found.
• What is the probability of randomly selecting 12
  cameras and finding no defects?

23
Quality Control

• We are looking for the probability of getting 12
  good cameras.
• P(12 good cameras)
• Since the defect rate was found to be 5, the P(1
  good camera) _____
• P(12 good cameras)
• P(good and good and good )
• P(all 12 good) 0.9512 0.540