1 counting problems from fred greenleaf’s qr book compiled by samuel marateck
TRANSCRIPT
3
Given an urn with 20 marbles 8 of which are
green, what is the probability of choosing
a green one?
5
This also means that if you perform the
experiment many times, let’s say 100
times, you’d expect to choose a green
marble 40 times.
7
What is the probability of choosing a non-
green marble?
P(non-green) = # of non-green marbles
total # of marbles
= 12/20 or 60%
9
Throw a pair of dice.What combinations can we get?1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
These are independent events in that the
result of one throw does not effect the
result of the next throw.
10
Which ones add to 7 or 11?
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
There are six that add to 7 and two that
add to 11.
There are 36 total outcomes.
11
What’s the probability of gettinga 7 or 11?
P(7 or 11) = # of 7 or 11
total # of outcomes
In probability when used for independent
Events, the or means use a “+”, so we add
the probabilities.
P(7 or 11) = P(7) + P(11) = 6/36 + 2/36 =8/36
12
A physicist says she will give you $3 if
you get a 7 or 11; but you will have to give
her $1 for any other outcome. Are these fair
odds?
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Let’s say you throw 36 times. You’d expect
to win 8x3 or 24 dollars but lose 36 – 8 or
28 times and have to pay $28. So you’d
loose $4 in 36 throws.
14
What are fair odds?
Let w be the amount you get if you win and
v be the amount you must pay if you lose.
So 8w = 28v or w/v = 28/8 = 3.5 .
If you give her $1 if you lose, she must
give you $3.50 if you win
15
How many two letter words can be formed
using the Roman alphabet, where any two
letters can form a word?
16
Let’s say you choose an A first. So you
can choose 26 letters as your second letter.
But you can choose 26 letters as your first
one. And for each of these you can chose
26 for your second one. So the total is
26x26.
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This is an example of a list with replacement
since you can use a letter again once it
has been used. It is also an ordered list
since the order is important. The word AB
is different from the word BA.
19
How many 5 letter words can you compose
If no letters are repeated?
This is an example of an ordered list
without replacement, since once a letter
has been used it cannot be used again.
20
For your first choice you have 26 letters.
For each of these letters you have 25
letters that have not been chosen. And for
each of these you have 24 that have not
been chosen and so on. So the answer
Is 26x25x24x23x22.
21
20 players are in a tournament. The top 5
players will be ranked, first, second, etc.
How many ways can they be ranked?
This is an example of an ordered list with
without replacement.
22
If player A is ranked first, then there are
19 players who can be ranked second.
And for each of these there are 18 who
can be ranked third and so forth.
24
You have a deck of 13 cards and you draw
a card from the deck. You then replace it.
What is the probability of drawing a different
card from the deck?
25
You have a deck of 13 cards and you draw
a card from the deck. You then replace it.
What is the probability of drawing a different
card from the deck?
Since there are 12 cards that have not been
drawn, the probability is 12/13 or 92%.
26
If three cards are drawn with replacement
from this deck, what is the probability they
will all be different?
27
If three cards are drawn with replacement
from this deck, what is the probability they
will all be different?
12/13 * 11/13 or 78%
28
If four cards are drawn with replacement
from this deck, what is the probability they
will all be different?
29
If four cards are drawn with replacement
from this deck, what is the probability they
will all be different?
12/13 * 11/13 * 10/13 or 60%
30
If five cards are drawn with replacement
from this deck, what is the probability they
will all be different?
31
If five cards are drawn with replacement
from this deck, what is the probability they
will all be different?
12/13 * 11/13 * 10/13 * 9/13 or 42%
32
If six cards are drawn with replacement
from this deck, what is the probability they
will all be different?
33
If six cards are drawn with replacement
from this deck, what is the probability they
will all be different?
12/13 * 11/13 * 10/13 * 9/13 * 8/13 or 26%
34
You have a deck of 5 cards and you draw
5 cards from the deck replacing them each
time.
What is the probability of drawing 5 different
card from the deck?
38
Combination Symbols
( n m) this is the symbol for n take m.How many committees of 3 can be chosenfrom 5 ?
This is an example of a unordered listwithout replacement. The order thatpeople appear on a committee is notimportant; but once a person is chosenhe cannot be chosen again.
40
If you write this with factorials,
you have to complete 5*4*3 with 2*1; but
this means you have to divide by 2*1. So
we get 5*4*3*2*1/(3*2*1 * 2*1) or
5!/(3! * 2!).
46
If you flip a coin 5 times, how many different
permutations can you get?
There are two choices for each coin, H or T.
So the number of permutations are 25, or
32.
How would you categorize this?
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It is an ordered list with replacement.
For instance, HTHTH is different from
THTHT. It is with replacement since
once a coin lands as a head, it can land
as a head again.
60
What do these add up to?
( 5 0) + ( 5 1) + ( 52) + ( 5 3) + ( 5 4) + ( 5 5)
1 + 5 + 10 + 10 + 5 + 1 = 32
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The probability of each throw is
( 5 0)/32 + ( 5 1)/32 + ( 52)/32 + ( 5 3)/32 +
( 5 4)/32 + ( 5 5)/32
1/32 +5/32+10/32+10/32+5/32+1/32 =32/32
Or 100%
62
If you choose two cards from a deck, what is
the probability that they are both aces?
How do you categorize this?
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This is an unordered list without
replacement, since once a card
has been drawn, it cannot be drawn again.
64
If you choose two cards from a deck, what is
the probability that they are both aces?
P(1 ace) = 4/52
P(2 aces)?
65
Since there are 3 aces left in the deck of51 cards,
P(2nd ace) = 3/51 and the P(2 aces) =
4/52*3/51
66
If you choose two cards from a deck, what is
the probability that they both have the
same face value?