Review Innovations
CE Review for Nov 2022 – Algebra 1
This simplified rule can be extended to any number of
mutually exclusive events.
PROBABILITY
For outcomes that are equally likely to occur
𝑃=
Number of favorable outcomes
Number of all possible outcomes
If the probability for an event to happen is 𝑃 and the
probability for it to fail is 𝑄, then 𝑃 + 𝑄 = 1.
Probability for Complementary Events
The complementary of event A is the event not A. Event
not A is usually denoted as A’ or Ac.
U
Conditional Probability
The probability that event B occurs given that event A has
occurred is denoted by 𝑃(𝐵 | 𝐴) . This is called the
conditional probability.
The Multiplication Rule (The AND Rule)
The probability that two events A and B will occur in
sequence is
𝑃(𝐴 and 𝐵) = 𝑃(𝐴) × 𝑃(𝐵 | 𝐴)
Baye’s Theorem
𝑃(𝐴) × 𝑃(𝐵 | 𝐴) = 𝑃(𝐵) × 𝑃(𝐴 | 𝐵)
A
𝑃(𝐴) = 1 − 𝑃(𝐴! )
Fundamental Principle of Counting
If event E1 can have n1 different outcomes, even E2 can have
n2 different outcomes, …, and event Em can have nm different
outcomes, then it follows that the number of possible
outcomes in which composite events E1, E2, …, Em can have
is
If events A and B are independent, then the rule can be
simplified to
This simplified rule can be extended for any number of
independent events.
In sets, the notation is
Permutation
An arrangement of objects in a definite order
The number of permutations on n different objects taken
r at a time is
𝑛!
𝑁 = 𝑛P𝑟 =
(𝑛 − 𝑟)!
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
A∩B
N = n1 × n2 × … × nm
We call this The Multiplication Principle.
P( A and B) = P( A) ´ P(B)
A
Ac
B
Combination
Differs from permutation in that it does not involve the
arrangement of objects nor the order of selection
The number of combinations of n objects taken r at a time
is
where ∩ = intersect
𝑁 = 𝑛C𝑟 =
The Addition Rule (The OR Rule)
The probability that events 𝐴 or 𝐵 will occur is given by
𝑃(𝐴 or 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 and 𝐵)
In sets, the notation is
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
Note:
nPn = n!
0! = 1
𝑛!
𝑟! (𝑛 − 𝑟)!
nCn = 1
nC0 = 1
Examples
1. What is the probability of drawing a Queen or a diamond
from a standard deck of cards?
Answer: 4/13
A∩B
A
B
A∪B
where ∪ = union
If events 𝐴 and 𝐵 are mutually exclusive, then the rule
can be simplified to
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
A
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A∪B
B
2. A bag contains 3 white and 5 red balls. If two balls are
drawn in succession without replacement, what is the
probability that…
2a. both balls are red?
Answer: 5/14
2b. both are white?
Answer: 3/28
2c. one is red and the other is white?
Answer: 15/28
2d. both have the same color?
Answer: 13/28
3. In a poker hand consisting of five cards, what is the
probability that…
3a. all 5 are diamonds?
Answer: 33/66,640
3b. all 5 are of the same suit?
Answer: 33/16,660
3c. 2 are clubs and 3 are hearts?
Ans: 143/16,660
4. In a family of five children, what is the chance that there
are three boys and two girls?
Answer: 5/16
Davao FB: Review Innovations Davao Branch
( (082) 221-1121 0930-256-0998
Review Innovations
CE Review for Nov 2022 – Algebra 1
5. Given a class of 12 girls and 10 boys. What is the
probability that a committee of five, chosen at random
from the class, consists of:
5a. only girls
Answer: 0.03
5b. At least one girl
Answer: 0.99
6. Throw two dice. What is the probability that the two
scores are different?
Answer: 5/6
9.
A certain statistical observation found that it has a
probability of failing once equal to 0.422, twice equal to
0.141 and thrice equal to 0.016. Determine the probability
that it will fall once or thrice.
Answer: 0.438
10. From a box containing 5 white balls, 7 blue balls, and 10
black balls, one ball is drawn at random. Determine the
probability that it is blue or white?
Answer: 6/11
Answer: 0.3452
11. A store has three kinds of toys given in every purchase.
What is the probability of getting all three toys in five
purchases?
Answer: 0.6173
8. In a room of 23 people, what is the probability that at least
2 people have the same birthday? Assume birthdays are
uniformly distributed across the year and there is no leap
year complication.
Answer: 0.50
12. In a population of 100,000 females, 89.835% can expect to
live to age 60, while 57.062% can expect to live to age 80.
Given that a woman is 60, what is the probability that she
lives to age 80?
Answer: 63.52%
9. Urn 1 contains 5 white balls and 7 black balls. Urn 2
contains 3 whites and 12 blacks. A fair coin is flipped. If
it shows heads, a ball from urn 1 is drawn; if it is tails, a
ball from urn 2 is drawn. Suppose a white ball was
selected. What is the probability that this ball was taken
from Urn 2?
Answer: 12/37
13. A bin contains 25 light bulbs, 5 of which are in good
condition and will function for at least 30 days of use, 10
of which are partially defective and will fail on the
second day of use and 10 of which are totally defective
and will not light up. Given that a randomly selected
bulb initially lights up, what is the probability that it will
still be working after one week?
Ans: 1/3
7. If six cards are selected at random from a standard deck,
what is the probability that there will be no pairs?
10. A truth serum given to a suspect is known to be 90%
reliable when the person is guilty and 99% reliable when
the person is innocent. If the suspect was selected from a
group of suspects of which only 5% have ever committed
a crime, and the serum indicates that he is guilty, what is
the probability that he is innocent?
Answer: 0.174
For Practice
1.
A face of a coin is either head or tail. If three coins are
tossed, what is the probability of getting three tails?
Answer: 1/8
2.
A coin is tossed twice. What is the probability that at
least one head occurs?
Answer: 3/4
3.
Suppose you toss a coin and then roll a die. What is the
probability of obtaining a tail and then rolling a 1?
Answer: 1/12
4.
5.
6.
A die is loaded in such a way that an even number is
twice as likely to occur as an odd number. Find the
probability that if this die is rolled, a number less than 4
occurs.
Answer: 4/9
The probability that a doctor correctly diagnoses a
particular illness is 0.7. Given that the doctor makes an
incorrect diagnosis, the probability that the patient
enters a lawsuit is 0.9. What is the probability that the
doctor makes an incorrect diagnosis, and the patient
sues?
Answer: 0.27
One bag contains 4 white balls and 3 black balls, and a
second bag contains 3 white balls and 5 black balls. One
ball is drawn at random from the second bag and is
placed unseen in the first bag. What is the probability
that a ball now drawn from the first bag is black?
Ans: 29/64
7.
The probability that a family owns a washing machine is
0.78 and the probability that it owns a VCR is 0.71. The
probability that it owns both a washing machine and a
VCR is 0.58. What is the probability that a randomly
selected family owns a washing machine or a VCR?
Ans: 0.910
8.
The probability that both stages of a two-stage missile
will function correctly is 0.95. The probability that the
first stage will function correctly is 0.98. What is the
probability that the second stage will function correctly
given that the first one does?
Answer: 0.969
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( (02) 8735-9161
0919-227-9194
14. The probability that an automobile being filled with
gasoline will also need an oil change is 0.25; the
probability that it needs a new oil filter is 0.40; and the
probability that both the oil and filter need changing is
0.14.
14a. If the oil had to be changed, what is the probability
that a new oil filter is needed?
Answer: 0.56
14b. If a new oil filter is needed, what is the probability
that the oil has to be changed?
Answer: 0.35
15. A box contains 5 defective and 195 non-defective cell
phones. A quality control engineer selects 2 cell phones
at random without replacement. What is the probability
that exactly 1 is defective?
Answer: 0.0490
16. A box of electronic items contains 16 which pass
international product standards and 4 which do not. If
three items are selected at random from the box, what is
the probability that all three pass international product
standards?
Answer: 0.4912
17. A random sample of 15 persons is taken from a
population in which 42% are Muslims. What is the
probability that exactly 6 persons in the sample are
Muslims?
Answer: 0.2041
18. A pump has a failure, on the average, once in every 5,000
hours of operations. What is the probability that no
failure will occur in 10,000 hours of operation?
Answer: 0.135
19. Four cards are drawn at random from a deck of 52 cards.
What is the probability that all four are face cards?
Answer: 99/54,145
20. The probabilities that three men hit a target are 1/6, 1/4,
and 1/3, respectively. Each shoot once at the target. If
only one of them hits the target, find the probability that
it was the first man.
Answer: 6/31
21. Studies have shown that 72.5% of apprehended
snatchers in Metro Manila are drug users. In a random
sample of 16 apprehended snatchers, what is the
probability that exactly 10 of them are drug users?
Answer: 0.139
22. Aedan tries to eat at least five pieces of fruit each day.
The probability that he does is 0.6, independent of any
other day. What is the probability that Aedan eats at
least five pieces of fruit on more than half the days in a
given week?
Answer: 71%
Davao FB: Review Innovations Davao Branch
( (082) 221-1121 0930-256-0998