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Elementary Statistics and
Inference
22S:025 or 7P:025
Lecture 18
1
Elementary Statistics and
Inference
22S:025 or 7P:025
Chapter 14
2
14. More About Chance
A.
Listing All Possibilities
Suppose two dice are thrown, the possible ways the
dice could fall is 36.
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1
14. More About Chance (cont.)
4
14. More About Chance (cont.)
„
Using this chart we can list the probabilities (chances) of
obtaining a total on the dice from 2 to 12.
X (Total)
P(X)
2
1/36
3
2/36
4
3/36
5
4/36
6
5/36
7
6/36
8
5/36
9
4/36
10
3/36
11
2/36
12
1/36
P( X ) =
Total ways of success
Total ways Event can occur
Sum=1.00
5
14. More About Chance (cont.)
„
Graphically
6/36
5/36
4/36
3/36
2/36
1/36
0
1
ƒ
2
3
4
5
6
7
8
9
10
11
12
If we had 3 dice, the number of possible ways
the dice could fall is 6 x 6 x 6 = 216.
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2
14. More About Chance (cont.)
B.
The Addition Rule
„
Two events (outcomes) are mutually exclusive when
the occurrence of one prevents the occurrence of the
second event – one excludes the other
other.
Example: If a deck is used to select a card, the
outcome can be a “heart” or a “spade” – they
are mutually exclusive.
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14. More About Chance (cont.)
„
If events are mutually exclusive, to find the probability
(chance) or either happening you add the probabilities.
P ( heart) + P (spade) =
13 13 26 1
+
=
=
52 52 52 2
Example: Someone throws a pair of dice. Is the chance
1 1 1
of at least one Ace + = ? See Figure 1
6 6 3
⎛ 1 1 1 11 ⎞
- ⎜ + − = ⎟.
⎝6
6
36
36 ⎠
ƒ If two events are not mutually exclusive, do not add the
chances – the sum is too big.
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14. More About Chance (cont.)
Exercise Set B (pp. 242-243) #2, 3, 4, 5
#5. (p.243) – A box contains 10 tickets numbered 1-10.
Five draws will be made at random with replacement
from this box
box. True or False: there are 5 chances in 10
of getting at least one “7” in the five draws.
False – the draws are not mutually exclusive
P (7) =
1
for each draw.
10
9
3
14. More About Chance (cont.)
#6. A number is drawn at random from a box. There is
a 20% chance for it to be 10 less, and a 10% chance
for it to be 50 or more. The chance of the number to
be between 10 and 50 – 70%.
20%
70%
10%
10
50
10
14. More About Chance (cont.)
C.
Two Frequently Asked Questions
Question: 1) What is difference between mutually
exclusive and independent?
2) When do I add probabilities, and when do
I multiply probabilities?
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14. More About Chance (cont.)
Answer:
Two events are mutually exclusive if the occurrence
of one prevents the other from happening.
‰ Two events are independent if the occurrence of
one does not change the chances (probability) of
the other.
‰ The addition rule finds the chance (probability) that
at least one of two things happens -- add
probabilities if events are mutually exclusive
‰ If need to determine probability that both
independent events happen – multiple the
probabilities. (for dependent events, the
multiplication rule uses conditional probabilities.)
‰
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4
14. More About Chance (cont.)
Example 6: p. 244
A die is rolled 6 times; a deck of cards is shuffled.
a) The chance the first roll is an ace or the
11 1 1 1
last roll is a six is 36 = 6 + 6 − 36
b) The chance that the first roll is an ace and the
1 1 1
= ×
last roll is an ace is
36
6 6
13
14. More About Chance (cont.)
c) The chance that the top card is the ace of
spades or the bottom card is the ace of
1
1
1
spades is 52 + 52 = 26
d) The chance the top card is an ace of
spades and the bottom card is an ace of
spades is 0
14
14. More About Chance (cont.)
p. 247 – Exercise 3
3. A deck of cards is shuffled. True or false, and explain
briefly:
a)
b)
c)
d)
e)
f)
The chance that the top card is the jack of clubs equals 1/52.
The chance that the bottom card is the jack of diamonds
equals
q
1/52.
The chance that the top card is the jack of clubs or the bottom
card is the jack of diamonds equals 2/52.
The chance that the top card is the jack of clubs or the bottom
card is the jack of clubs equals 2/52.
The chance that the top card is the jack of clubs and the
bottom card is the jack of diamonds equals 1/52 x 1/52.
The chance that the top card is the jack of clubs and the
bottom card is the jack of clubs equals 1/52 x 1/52.
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5
14. More About Chance (cont.)
p.247 – Exercise 4
4. The unconditional probability of event A is ½. The
unconditional probability of event B is 1/3. Say whether
each of the following is true or false, and explain briefly.
a)
b)
c)
d)
e)
f)
The chance that A and B both happen must be ½ x 1/3 = 1/6.
If A and B are independent,
p
the chance that they
y both
happen must be ½ x 1/3 = 1/6.
If A and B are mutually exclusive, the chance that they both
happen must be ½ x 1/3 = 1/6.
The chance that at least one of A or B happens must be ½ +
1/3 = 5/6.
If A and B are independent, the chance that at least one of
them happens must be ½ + 1/3 = 5/6.
If A and B are mutually exclusive, the chance that at least
one of them happens must be ½ + 1/3 = 5/6.
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14. More About Chance (cont.)
Exercise Set D – p. 250
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14. More About Chance (cont.)
Exercise Set D – p. 250
4.
a) A die is rolled 3 times. What is the
chance of getting at least one ace?
b) Same, with 6 rolls?
c) Same, with 12 rolls?
5.
A pair of dice is rolled 36 times. What is the chance of
obtaining at least one double-ace?
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6
14. More About Chance (cont.)
Review Exercises – pp. 252-253 #1 – 10
When a die is rolled, each of the 6 faces is equally
likely to come up. A deck of cards has 4 suits (clubs,
diamonds, hearts, spades) with 13 cards in each suit –
2 3
2,
3, …, 10
10, jack
jack, queen
queen, king
king, ace
ace. See pp
pp. 222 and
226.
1. A pair of dice are thrown.
a) Find the chance that both dice show 3 spots.
b) Find the chance that both dice show the same number
of spots.
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20
14. More About Chance (cont.)
2. In the game of Monopoly, a player rolls two dice,
counts the total number of spots, and moves that many
squares. Find the chance that the player moves 11
squares (no more and no less.
3. True or false, and explain:
a) If a die is rolled three times, the chance of getting at
least one ace is 1/6 + 1/6 + 1/6 = ½.
b) If a coin is tossed twice, the chance of getting at least
one head is 100%.
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7
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14. More About Chance (cont.)
4.
Two cards will be dealt off the top of a well-shuffled
deck. You have a choice:
i.
ii
ii.
To win $1 if at least one of the two cards is a
queen.
T win
To
i $1 if th
the fifirstt iis a queen.
Which option is better? Or are they equivalent?
Explain.
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14. More About Chance (cont.)
5. The chance of A is 1/3; the chance of B is 1/10. True
or false, and explain:
a) If A and B are independent, they must also be mutually
exclusive
exclusive.
b) If A and B are mutually exclusive, they cannot be
independent.
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8
14. More About Chance (cont.)
6.
One event has chance ½, another has chance 1/3.
Fill in the blanks, using one phrase from each pair
below, to make up two true sentences. Write out both
sentences.
“If
If you want to find the chance that
(i)
will
happen, check to see if they are (ii) . If so, you
can (iii) the chances.”
i.
ii.
iii.
at least one of the two events, both events
independent, mutually exclusive
add, multiply
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14. More About Chance (cont.)
7. Four draws are going to be made at random with
replacement from the box
1
2
2
3
3
Find the chance that “2” is drawn at least one time.
8. Repeat exercise 7, if the draws are made at random
without replacement.
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14. More About Chance (cont.)
9. One ticket will be drawn at random from each of the
two boxes shown below:
(A)
a)
b)
c)
1
2
3
(B)
1
2
3
4
Find the chance that:
The number drawn from A is larger than the one from
B.
The number drawn from A equals the one from B.
The number drawn from A is smaller than the one from
B.
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9
14. More About Chance (cont.)
10. There are two options:
i.
ii.
A die will be rolled 60 times. Each time it shows an
ace or a six, you win $1; on the other rolls, you win
nothing.
Sixty draws will be made at random with replacement
from the box
1
1
1
0
0
0
On each draw, you will be paid the amount shown on
the ticket, in dollars. Which option is better? Or are
they the same? Explain briefly.
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