Transcript probability_of_and_or_complements

Probability using And,Or and
Complements
Independent Events
• Two events are Independent if the
occurrence of 1 has no effect on the
occurrence of the other. (a coin tossed 2
times, the first toss has no effect on the
2nd toss)
• If A & B are independent events then the
probability that both A & B occur is:
• P(A and B) = P(A) • P(B)
•
A number cube is rolled and a coin is
tossed. Find the probabilities:
1. P(5)
2. P(heads)
3. P(5 and heads)
1. P(5) = 1/6
2. P(heads) = 1/2
3. P(5 and heads) = P(5)  P(heads)
=
1 1

6 2
=
1
12
Dependent Events
• Two events A and B are
dependent events if the
occurrence of one affects the
occurrence of the other.
Dependent Events
• If A & B are dependent events, then the
probability that both A & B occur is:
• P(A&B) = P(A) * P(B/A)
• The probability that B will occur given that
A has occurred is called the conditional
probability of B given A and is written
P(B|A).
Comparing Dependent and
Independent Events
• You randomly select two cards from a
standard 52-card deck. What is the
probability that the first card is not a face
card (a king, queen, or jack) and the
second card is a face card if
• (1) you replace the first card before
selecting the second, and
• (2) you do not replace the first card?
• (1) If you replace the first card before
selecting the second card, then A and B are
independent events. So, the probability is:
• P(A and B) = P(A) • P(B) = 40 * 12 = 30
52 52 169
• ≈ 0.178
• (2) If you do not replace the first card before
selecting the second card, then A and B are
dependent events. So, the probability is:
• P(A and B) = P(A) • P(B|A) = 40*12 = 40
52 51 221
• ≈ .0181
Mutually Exclusive Events
Intersection of A & B
• To find P(A or B) you must consider what
outcomes, if any, are in the intersection of
A and B.
• If there are none, then A and B are
mutually exclusive events and
P(A or B) = P(A)+P(B)
• If A and B are not mutually exclusive, then
the outcomes in the intersection (A and B)
are counted twice when P(A) and P(B) are
added.
• So P(A and B) must be subtracted once
from the sum
EXAMPLE 1
• One six-sided die is rolled.
• What is the probability of rolling a multiple
of 3 or 5?
• P(A or B) = P(A) + P(B) = 2/6 + 1/6 = 1/2
• 0.5
EXAMPLE 2
• One six-sided die is rolled. What is the
probability of rolling a multiple of 3 or a
multiple of 2?
• A = Mult 3 = 2 outcomes (3,6)
• B = mult 2 = 3 outcomes (2,4,6)
• P(A or B) = P(A) + P(B) – P(A and B)
• P(A or B) = 2/6 + 3/6 – 1/6 =
• 2/3 ≈ 0.67
EXAMPLE 3
• In a poll of high school juniors, 6 boys took
French and 8 girls took french,11 boys
took math class and 7 girls took math.
• How many juniors surveyed were either
girl or took math?
•
•
•
•
•
A = girl
B = took math
P(A) = 15/32, P(B) = 18/32
P(A or B) = P(A) + P(B) – P(A and B)
P(A or B) = 15/32 + 18/32 – 7/32
= 26/32
= 13/16
Using complements to find
Probability
• The event A’, called the complement of
event A, consists of all outcomes that are
not in A.
• The notation A’ is read ‘A prime’.
Probability of the complement of an
event
• The probability of the complement
of A is :
• P(A’) = 1 - P(A)
EXAMPLE 4
• A card is randomly selected
from a standard deck of 52
cards.
• Find the probability of the
given event.
• a. The card is not a king.
1 – P(king) = 1 – 4/52
= 48/52 ≈ 0.923
• b. The card is not an ace or a jack.
• P(not ace or jack)
• 1 – P(ace or jack)= 1- P(4/52 + 4/52)
= 1- 8/52
= 44/52 ≈ 0.846
• In a survey of 200 pet owners, 103 owned dogs, 88 owned
cats, 25 owned birds, and 18 owned reptiles.
• 1. None of the respondents owned both a cat and a bird.
• What is the probability that they owned a cat or a bird?
• 113/200
• = 0.565
• 2. Of the respondents, 52 owned both a cat and a dog.
• What is the probability that a respondent owned a cat or a
dog?
• 139/200
• = 0.695
Assignment
Worksheet on Probability #’s 1-24