SOUTHERN BICOL COLLEGES
MELISSA G. VELASCO
BSED-III (MAJOR IN MATHEMATICS)
ENGR. MARIA ROMINA PRAC ANGUSTIA
TABLE OF CONTENTS . . . . . . ii
HISTORY OF PROBABILITY . . . . . 1
PROBABILITY . . . . . . . 2
COMPLEMENTARY PROBABILITY . . . . 3
JOINT PROBABILITY . . . . . . 4-5
CONDITIONAL PROBABILITY . . . . . 6
INDEPENDENT PROBABILITY . . . . . 6
REPEATED TRIALS PROBABILITY . . . . 7
HISTORY OF PROBABILITY
A gambler’s dispute in 1654 led to the creation of a mathematical
theory of probability by two famous French mathematicians, Blaise Pascal
and Pierre de Fermat. Chevalier de Mere, a French nobleman had an
interest about the gambling so he question the two famous mathematician
about the scoring of the game. This problem and others posed by de Mere
led to an exchange of letters between Pascal and Fermat in which the
fundamental principles of probability theory were formulated for the first
GAMING / GAMBLING
The game consisted of throwing a pair of dice 24 times; the scoring
was to decide whether or not to bet even money on the occurrence of at
least one “double six” during the 24 throws.
Is the measure of how likely an event is to occur.
𝐸𝑉𝐸𝑁𝑇𝑆(𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑟𝑒𝑠𝑢𝑙𝑡 𝑜𝑓 𝑎 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡)
𝑂𝑈𝑇𝐶𝑂𝑀𝐸𝑆( 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑟𝑒𝑠𝑢𝑙𝑡 𝑜𝑓 𝑎 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡)
What is the probability of getting head when flipping a coin?
PROBABILITY NUMBER LINE
Impossible unlikely equal chances likely certain
0 0.5 1
0% 50% 100%
Is not of that event.
The sum of the probability is equal to 1.
P(A) + P(B) = 1 or P(not A) =1 – P(A)
A number is chosen at random from a set of whole numbers from 1 to 150.
Calculate the probability that the chosen number is not a perfect square.
P(not perfect square)=1 – P(perfect square)
= 0.92 or 92%
Is the likelihood of more than one event occurring at the same time.
2 TYPES OF JOINT PROBABILITY
Mutually Exclusive Event-(without common outcomes)
Non-Mutually Exclusive Event-(with common outcomes)
Mutually exclusive event
P(A and B)=0
P(A or B)=P(A) + P(B)
Non-mutually exclusive event
P(A or B)=P(A) + P(B) – P(A∩B)
EXAMPLE: Mutually exclusive event
A group of teacher is donating blood during blood drive. A student has a
probability of having a type O blood and a
probability of having a type A blood. What
is the probability that a teacher has type A or O type of blood?
P(AUB)=P(A) + P(B)
= 0.6074 or 60.74%
EXAMPLE: Non-mutually exclusive event
What is the probability when you shuffle a deck of cards to get a black card or a 7
P(black card or 7 card)=P(BC) + P(7C) – P(BC Ո 7C)
= 0.5384 or 53.84%
Is the probability of an event (A), given that another event (B) has already
P(A and B)=P(A) × P(B/A) or P(A/B)=
P(B and A)
Where: P(A)=prob. Of event A may happen.
P(B)=prob. Of event B may happen.
P(B/A)=prob. of event B given A.
EXAMPLE: dependent event
70% of your friends like Chocolate, and 35% like Chocolate and like Strawberry.
What percent of those who like Chocolate also like Strawberry?
= 0.5 or 50%
EXAMPLE: independent event
2 boxes contain of small balls with different color. In the 1st box 2 yellow, 5 red.
2nd box 3 blue, 4 white. What is the probability of getting a red and a white.
Formula and solution:
P(A and B) = P(A) × P(B)
P(R∩W) = P(R) × P(W)
=0.4081 or 40.81%
REPEATED TRIALS PROBABILITY (binomial)
Probability that an event will occur exactly “r” times out of “n” trials.
Also called Bernoulli trial
Binomial trial-is the random experiment with exactly two possible
outcomes. (success & failure)
Jacob Bernoulli a Swiss mathematician in 17th century.
P=nCr (𝑝) 𝑟
p = prob. of success
q = prob. of failure
n = no. of trials
r = successful outcomes
A die is thrown 6 times. If getting an odd number is a success, what is the
probability of 5 successes?
n = 6
r = 5
= 0.0937 or 9.37%