464
8.2. GEOMETRIC SERIES
turn. Explain why the probability of this event is
5
6
5
6
1
6
=
5
6
2 1
6
.
(c) Now suppose you fail to roll the first six on your second turn. Explain why the
probability is
5
6
5
6
5
6
5
6
1
6
=
5
6
4 1
6
that you to roll the first six on your third turn.
(d) The probability of you rolling the first six is the probability that you roll the
first six on your first turn plus the probability that you roll the first six on your
second turn plus the probability that your roll the first six on your third turn,
and so on. Explain why this probability is
1
6
+
5
6
2 1
6
+
5
6
4 1
6
+ · · · .
Find the sum of this series and determine the probability that you roll the first
six.
4. The goal of a federal government stimulus package is to positively affect the economy.
Economists and politicians quote numbers like “k million jobs and a net stimulus to
the economy of n billion of dollars.” Where do they get these numbers? Let’s consider
one aspect of a stimulus package: tax cuts. Economists understand that tax cuts or
rebates can result in long-term spending that is many times the amount of the rebate.
For example, assume that for a typical person, 75% of her entire income is spent (that
is, put back into the economy). Further, assume the government provides a tax cut or
rebate that totals P dollars for each person.
(a) The tax cut of P dollars is income for its recipient. How much of this tax cut
will be spent?
(b) In this simple model, we will say that the spent portion of the tax cut/rebate
from part (a) then becomes income for another person who, in turn, spends
75% of this income. After this “second round" of spent income, how many
total dollars have been added to the economy as a result of the original tax
cut/rebate?
(c) This second round of spending becomes income for another group who spend
75% of this income, and so on. In economics this is called the multiplier effect.
Explain why an original tax cut/rebate of P dollars will result in multiplied
spending of
0.75P(1 + 0.75 + 0.75
2 + · · · ).
dollars.
8.2. GEOMETRIC SERIES
turn. Explain why the probability of this event is
5
6
5
6
1
6
=
5
6
2 1
6
.
(c) Now suppose you fail to roll the first six on your second turn. Explain why the
probability is
5
6
5
6
5
6
5
6
1
6
=
5
6
4 1
6
that you to roll the first six on your third turn.
(d) The probability of you rolling the first six is the probability that you roll the
first six on your first turn plus the probability that you roll the first six on your
second turn plus the probability that your roll the first six on your third turn,
and so on. Explain why this probability is
1
6
+
5
6
2 1
6
+
5
6
4 1
6
+ · · · .
Find the sum of this series and determine the probability that you roll the first
six.
4. The goal of a federal government stimulus package is to positively affect the economy.
Economists and politicians quote numbers like “k million jobs and a net stimulus to
the economy of n billion of dollars.” Where do they get these numbers? Let’s consider
one aspect of a stimulus package: tax cuts. Economists understand that tax cuts or
rebates can result in long-term spending that is many times the amount of the rebate.
For example, assume that for a typical person, 75% of her entire income is spent (that
is, put back into the economy). Further, assume the government provides a tax cut or
rebate that totals P dollars for each person.
(a) The tax cut of P dollars is income for its recipient. How much of this tax cut
will be spent?
(b) In this simple model, we will say that the spent portion of the tax cut/rebate
from part (a) then becomes income for another person who, in turn, spends
75% of this income. After this “second round" of spent income, how many
total dollars have been added to the economy as a result of the original tax
cut/rebate?
(c) This second round of spending becomes income for another group who spend
75% of this income, and so on. In economics this is called the multiplier effect.
Explain why an original tax cut/rebate of P dollars will result in multiplied
spending of
0.75P(1 + 0.75 + 0.75
2 + · · · ).
dollars.