Section 7.1: Chance and Probability

1
Section 7.1 Chance and Probability
2
Probability

• Chance behavior is unpredictable in the short run
  (you probably cant guess what number will be
  drawn, which side the coin will fall on, etc.),
  but chance behavior has a regular and predictable
  pattern in the long run.
• What does that mean?

3
Probability of tossing heads
4
Probability

• With a coin toss, probability 0.5 means occurs
  half the time in a very large number of trials.
• The idea of probability is empirical
• based on data rather than theorizing.

5
Probability

• A probability experiment is a trial through which
  specific results (counts or measurements) are
  obtained.
• The result of a single trial is an outcome.
• The set of all possible outcomes is the sample
  space.
• An event consists one or more outcomes and is a
  subset of the sample space. If it has only one
  outcome, it is referred to as a simple event.

6
Example

• For quality control, you randomly select a
  computer chip from a batch that has been
  manufactured that day. Event A is selecting a
  defective chip.
• The experiment is selecting the chip to inspect.
• The possible outcomes are defective or not
  defective, which together make up the sample
  space.
• Since there is only one possible outcome, it is a
  simple event.

7
Probability Theory

• Mathematical study of randomness.
• Originated by Pierre de Fermat and Blaise Pascal
• People wanted to see the fair value of bets
  (how should I bet).

8
Law of Averages

• Its a myth.
• People think if the same outcome occurs over and
  over, a different outcome will occur next (heads
  5 times in a rowit must be tails next.)
• Because flipping a coin each time has no impact
  on the next flip (coins dont have memories),
  these events are mutually exclusive (they have no
  impact on each other)

9
Law of Averages
10
An example

• A few years ago, Dear Abby published in her
  advice column a letter from a distraught mother
  of 8 girls. It seems that she and her husband
  had planned to limit their family to four
  children. When all four were girls, they tried
  againand again, and again. After 7 straight
  girls, even her doctor had assured her that the
  law of averages was in our favor 100 to 1.
  Unfortunately for this couple, having children is
  like tossing coins. Eight girls in a row is
  highly unlikely, but once 7 girls have been born,
  it is not at all unlikely that the next child
  will be a girland it was.

11
Was it chance or causation?

• In 1979, 2 of the 8 town wells serving Woburn,
  MA, were found to be contaminated with organic
  chemicals. Alarmed citizens began counting
  cancer cases. Between 1964 and 1983, 20 cases of
  childhood leukemia were reported. This is an
  unusual number of cases for this rather rare
  disease. The residents believed the well water
  had caused the leukemia and proceeded to sue the
  2 companies held liable for the contamination.

12
The results of an investigation

• Investigators obtained complete data on everyone
  who had lived in the neighborhoods during the
  periods in question and to estimate their
  exposure to the suspect drinking water.
• They looked for other factors such as
  occupational exposure to toxins and smoking.
• The verdict There was evidence of an
  association between drinking water from the 2
  wells and developing childhood leukemia.

13
(No Transcript)
14
Personal Probability

• Personal Judgment (how likely something is to
  happen based on your opinion).
• We base our decisions on them.
• Because they are opinions, they arent right or
  wrong.
• Still a number between 0 and 1.

15
Probability is used to describe risk

• We feel safer when risk seems under our control
  (risk while driving, as opposed to risk of
  asbestos or terrorism).
• Its hard to comprehend very small probabilities
  (we overestimate small risks and underestimate
  large risks).
• Probabilities for risks like asbestos are only
  estimates (we should suspect that experts have
  underestimated the risk).

16
Work

• Group Work
• White Book Page 125, 1-10
• Homework
• Green Book Pages 411..422 4, 8, 12-14, 18