UNIT 5: PROBABILITY
Sample Space
Rock, Paper,
Scissors
Tournament
History
• As old as civilization.
• Egyptians used a small mammal bone as a
4 sided die (3500 BC)
• Games of chance were common in Greek
& Roman times
• Formal study – began with Blaise Pascal &
Pierre de Fermat due to gambling
problems.
Activity (1 in 6)
Roll your die seven times to imitate the process of the
seven friends buying their soda. How many of them
won a prize.
2. Repeat step 1 four more times. In the 5 repetitions,
how many times did three or more of the group win a
prize?
3. Combine results with your classmates. What percent of
the time did the friends come away with three or more
prizes, just by chance?
4. Based on your answer in step 3, does it seem plausible
that the company is telling the truth, but that the seven
friends just got lucky?
1.
Probability
• It’s the long-run relative frequency.
Empirical Probability
• This is the relative frequency that is
obtained by actually doing the experiment.
Activity: Coin Toss Graph
Law of Large Numbers
• With more repetitions, the
empirical probability gets closer to
the theoretical probability.
• Theoretical Probability is based on
what I should get (in
theory)….based on the sample
space.
Probability
• It is the outcome of a chance process that is a number
between 0 and 1 that describes the proportion of times
the outcome would occur in a very long series of
repetitions.
Life Insurance:
We can’t predict whether a particular person will die
in the next year. But, the National Center for Health
Statistics says that the proportion of men aged 20
to 24 years who die in any one year is 0.0015.
*What percent of men (20 to 24) will die?
*The probability of a woman (20 to 24) will die is one-third
that of a man. What percent of women (20 to 24) will die?
*Would you be surprised to learn that they charge a man (20
to 24) three times more than a woman (20 to 24)?
According to the “Book of Odds,” the
probability that a randomly selected U.S.
Adult usually eats breakfast is 0.61.
• What does the probability 0.61 mean in this setting?
• Does this mean that if 100 U.S. adults are chosen at
random, exactly 61 of them usually eat breakfast?
Match the following probabilities with each
statement…
0
0.01
0.3
0.6
0.99
1
• This outcome is impossible…it can never occur.
• This outcome is certain. It will occur on every trial.
• This outcome is very unlikely, but it will occur once in a
while in a long sequence of trials.
• This outcome will occur more often than not.
Which of the following outcomes is
more probable?
HTHTTH
TTTHHH
Myth of Short-run Regularity
• Probability is predictable in the long run…not the
short run.
Myth of the “Law of Averages”
• Things do even out…in the long run.
• If I tossed a coin and got TTTTTT, which
of the following is more likely on the next
toss?
• Getting Heads
• Getting Tails
Simulation
• It is an imitation of chance behavior,
based on a model that accurately
reflects the situation.
• This is what we did in the 1 in 6 game
earlier
At a local high school, 95 students have permission to park on campus.
Each month, the student council holds a “golden ticket parking lottery” at
a school assembly. The two lucky winners are given reserved parking
spots next to the school’s main entrance. Last month, the winning
tickets were drawn by a student council member from the AP Stats
class. When both golden tickets went to members of that same class,
some people thought the lottery had been rigged. There are 28
students in the AP Stats class, all of whom are eligible to park on
campus. Design and carry out a simulation to decide whether it’s
plausible that the lottery was carried out fairly.
Let’s start at line 139 of the Random number table. Let the students
who can park be represented as 01 to 95 and the students in the AP
stats is 01 to 28. We will simulate drawing two tickets. We will use 18
trials.
55588
99404
70708
56934
48394
51719
41098
43563
In an attempt to increase sales, a breakfast cereal company
decides to offer a NASCAR promotion. Each box of cereal will
contain a collectible card featuring one of these NASCAR drivers:
Jeff Gordon, Dale, Earnhardt, Jr., Tony Stewart, Danica Patrick, or
Jimmie Johnson. The company says that each of the 5 cards is
equally kiely to appear in any box of cereal. A NASCAR fan
decides to keep buying boxes of the cereal until she has all 5
drivers’ cards. She is surprised when it takes her 23 boxes to get
the full set of cards. Should she be surprised? Design and carry
out a simulation to help answer this question.
Let 1-5 represent finding each of the five NASCAR drivers.
Since we want a full deck, then we will go until we get all 5
cards. We will keep up with the number of boxes that we
had to check in order to get all 5. Let’s start with line116.
14459
26056
31424
80371
65103
62253
50490
61181
38167
98532
62183
70632
23417
26185
41448
75532
Activity: Is it Fair?
Homework
• Page 293 (1, 3, 7, 9, 13, 17, 19, 23, 25)