Mar. 1 Statistic for the day: Number of years before the sequence of Easter dates repeats itself: 5,700,000

1
Mar. 1 Statistic for the dayNumber of years
before the sequence of Easter dates repeats
itself 5,700,000
Source http//webexhibits.org/calendars/

• Assignment
• Review probability topics, exercises
• in Chapters 15 and 16

These slides were created by Tom Hettmansperger
and in some cases modified by David Hunter
2
ExpectationInsurance
Example 14 p267 extended.
Suppose my insurance company has 10,000 policy
holders and they are all skateboarders. I
collect a 500 premium each year. I pay off
1500 for a claim of a skate board
accident. From past experience I know 10 will
file a claim. How much do I expect to make per
customer?
3
Pr(claim) .10 loss is 1500 - 500 1000
recorded as
-1000 Pr(no claim) .90 gain is
500 ---------------------------------------------
----------------------------- Expected value
.10x(-1000) .90x(500)
-100 450 350
dollars per customer -----------------------------
--------------------------------------------- Expe
cted value for the 10,000 customers
10,000x350
3,500,000 dollars per year
4
Alternatively Expect 10 or 1,000 claims for
1000x(-1000)
-1,000,000 loss Expect 90 or 9,000
earning 500 each or 4,500,000 gain So we
expect 4,500,000 - 1,000,000 3,500,000 net
profit
5
Some old ideas from chapter 7 (histograms) and
chapter 8 (bell shaped curves).
The standard deviation is roughly (3,600,000
3,400,000)/4 50,000 Actual value is 45,000
computed from a formula.
6
So 95 of the time the net profit will be
between 3,410,00 and 3,590,000 with expected
value 3,500,000
7
Craps

• Win on first roll if 7 or 11
• Lose on first roll if 2, 3, or 12
• Anything else on first roll becomes point
• Win on subsequent roll if point
• Lose on subsequent roll if 7
• Probability of winning 244/495, or 49.3

8
Betting ten dollars on a game of craps
You win 10 with probability .493 You lose 10
with probability .507 Expected winnings
.493(10) .507(-10) -.14
How is a fourteen-cent-loss relevant to your
situation if you only play once?
How is a fourteen-cent-gain relevant to your
situation if you are the casino and you play ten
thousand times a day?
9
What would a 95 interval be?
About -500 to 3500
Expected 1400
Std Dev About 1000
10
What if the casino plays 100,000 times a day?
Note Now all values are positive
11
Exercise 25 p276
Suppose 72 of children live with both
parents, 22 live with mother only, 3 live with
father only, 3 live with neither mother nor
father. Pr(live with 2 parents) .72 Pr(live
with 1 parent) .22 .03 .25 Pr(live with 0
parents) .03 Expected number of parents lived
with 2x.72 1x.25 0x.03 1.44 .25 0
1.69 parents If we sampled 1000 children and
counted the total number of parents, we would
expect around 1690 parents.
12
Calibrating personal probabilities of experts p.
290
13
Anchoring

• Do you think the number of cancer deaths in 1993
• was above or below 200,000?
• What do you think the number of cancer deaths was
• in 1993?

Answer 538,000

• Was the number of deaths due to aids
• in 1993 lower or higher than 30,000?
• What do you think the number of deaths due
• to aids in 1993 was?

Answer 16,885
14
Recall earlier quiz we didnt have

• Mary likes earrings and spends time at festivals
  shopping
• for jewelry. Her boy friend and several of her
  close girl
• friends have tattoos. They have encouraged her
  to also
• get a tattoo.
• Unknown to you, Mary will be sitting next to you
  in the
• next stat100.2 class.
• Which of the following do you think is more
  likely and why?
• Mary is a physics major.
• Mary is a physics major with pierced ears.

15
An answer of B (Mary is a physics major with
pierced ears) is impossible and illustrates the
Conjunction fallacy assigning higher
probability to a detailed scenario involving the
conjunction of events than to one of the simple
events that make up the conjunction.

A possible cause of this fallacy is
the Representative heuristic leads people to
assign higher probabilities than are warranted to
scenarios that are representative of how we
imagine things would happen.

16
Forgotten base rates

• Kahneman and Tversky example, p. 286
• Another possible example

Suppose that a dreaded disease affects 1 of
those who get tested for it. Also suppose that
the test is 99 accurate. What would you advise
a patient who tests positive if the test result
were the only piece of information?
True probability of disease about 50
17
Birthdays
How many people must be in the same room to
guarantee that at least 2 people will have the
same birthday?
Answer 366 (not allowing for leap years)
Different question How many people must be in
the same room to so that at least 2 people will
have the same birthday with a high probability?
18
Suppose there are 15 people in a room. What
is the probability that at least 2 people have
the same birthday?
Pr(at least one match) 1 - .75 .25
19
number in room Pr(at least one match)
15 .25
20 .41
23 .51
30 .71
40 .89
50 .97
60 .994