Elementary Statistics and
Inference
22S:025 or 7P:025
Lecture 20
1
Elementary Statistics and
Inference
22S:025 or 7P:025
Chapter 16 (cont.)
2
16 – The Law of Averages (cont.)
D.
Making a Box Model
ƒ Key Questions regarding box
‰What numbers go into making the box?
‰How many of each kind of number?
‰How many draws taken from the box?
3
1
16 – The Law of Averages (cont.)
ƒ Authors illustrate the
Box Model with a
Roulette Wheel – used
in gambling casino.
4
16 – The Law of Averages (cont.)
Facts About Roulette Wheel
38 pockets where ball can land
0, 00, 1-36
„
ƒ
0 and 00 – are green in color
ƒ
18 red, 18 black numbered pockets
ƒ
If you bet $1.00 on “red”, and a “red number is where the
ball stops – you win $1.00 plus your original dollar –
otherwise you lose your dollar.
5
16 – The Law of Averages (cont.)
18
R
18
B
2
G
ƒ
Chance of red is 18/38 = .47.
Chance of non
non-red
red is 20/38 = .53.
53
ƒ
The advantage is with the casino.
ƒ
Play the game 10 times – and bet on red each time,
what would be expected gain?
6
2
16 – The Law of Averages (cont.)
Note: This is like 10 draws from a box made with
replacement from the box with 38 tickets.
18
ƒ
+$1.00
20
-$1.00
Outcome:
RRRBG RRBBR
Win-Loss:
1 1 1 -1 -1 1 1 -1 -1 1
Net Gain:
1 2 3 2 1 2 3 2 1 2
In the next chapter we will discuss what would be
expected gain if you kept playing the game.
7
16 – The Law of Averages (cont.)
See Example 1 – page 283
„
Bet on a single number – win – gain $35 which gamblers
say is odds of winning are 35 to 1.
„
Suppose you play game 100 times and bet $1.00 on a
single number (say 17) each time – box model
35
1
37
-1
Expected gain after 100 plays:
−2
⎛ 1 ⎞ ⎛ 37 ⎞
35⎜ ⎟ + ⎜ ⎟(− 1) =
= $.0526 / play
38
⎝ 38 ⎠ ⎝ 38 ⎠
Exercise Set C – (pp. 284-285) #1, 2, 3
„
8
9
3
10
16 – The Law of Averages (cont.)
2.
A gambler plays roulette 25 times, putting a dollar on a
“split” each time (i.e., on line between 11 and 12 – so
he wins if either an 11 or 12 pocket receives the ball).
If he wins he gets $17
$17.00
00 plus the $1
$1.00
00 he bet
bet.
P ( win ) =
2
38
P (lose) =
36
38
11
16 – The Law of Averages (cont.)
His Net gain is like sum of draws from one of the boxes
below. Which one?
(i.)
0
00
36 unique tickets
(ii.)
2
17
34
-1
2
17
36
-1
(iii.)
12
4
16 – The Law of Averages (cont.)
3.
In one version of chuck-a-buck, 3 dice are rolled out of
a cage (total of 216 ways dice can show). You can bet
that all 3 show a given number, like “6”. The casino
pays 36 to 1, the better has 1 chance in 216 to win.
play
y 10 times,, betting
g$
$1.00 each time. The net
You p
gain is like the sum of 10 draws made at random with
replacement from the box:
1
36
215
-1
P (W ) =
In a single play – net gain would be
1
216
36 (− 215)
+
= $ − .83
216
216
13
16 – The Law of Averages (cont.)
E.
Review Exercises – pp. 285-286
3. A gambler loses ten times running at roulette. He
decides to continue playing because he is due for a
win, by the law of averages. A bystander advises
him to quit, on the grounds that his luck is cold.
Who is right? Or are both of them wrong?
Quit – expect to lose $-.05 on each play.
14
16 – The Law of Averages (cont.)
4. (a) A die will be rolled some number of times, and
you win $1 if it shows an ace more than 20%
of the time. Which is better: 60 rolls, or 600 rolls?
Explain.
(b) As in (a), but you win the dollar if the percentage
of aces is more than 15%.
(c) As in (a), but you win the dollar if the percentage
of aces is between 15% and 20%.
(d) As in (a), but you win the dollar if the percentage
of aces is exactly 16 2/3%.
15
5
16 – The Law of Averages (cont.)
7. A quiz has 25 multiple choice questions. Each question
has 5 possible answers, one of which is correct. A
correct answer is worth 4 points, but a point is taken off
for each incorrect answer. A student answers all the
questions by guessing at random. The score will be like
the sum of
draws from the box
. Fill in the blank with a number and the second with
a box of tickets. Explain your answers.
16
16 – The Law of Averages (cont.)
8. A gambler will play roulette 50 times, betting a dollar on
four joining numbers each time (like 23, 24, 26, 27 in
figure 3, p. 282). If one of these four numbers comes up,
she gets the dollar back, together with winnings of $8. If
p, she loses the dollar. So
anyy other number comes up,
this bet pays 8 to 1, and there are 4 chances in 38 of
winning. Her net gain in 50 plays is like the sum of
draws from the box
. Fill in the blanks;
explain.
17
16 – The Law of Averages (cont.)
10. Two hundred draws will be made at random with
replacement from the box
-3
-2
-1
0
1
2
3
(a) If the sum of the 200 numbers drawn is 30, what is
their average?
(b) If the sum of the 200 numbers drawn is -20,
20, what is
their average?
(c) In general, how can you figure the average of the 200
draws, if you are told their sum?
(d) There are two alternatives:
(i) Winning $1 if the sum of the 200 numbers drawn
is between -5 and +5.
(ii) Winning $1 if the average of the 200 numbers drawn
is between -0.025 and +0.025.
Which is better, or are they the same? Explain.
18
6
19
20
21
7