SOUTHERN BICOL COLLEGES

MASBATE CITY

COMPILATIONOF

DIFFERENTKINDS

OFPROBABILITY

MELISSA G. VELASCO

BSED-III (MAJOR IN MATHEMATICS)

ENGR. MARIA ROMINA PRAC ANGUSTIA

INSTRUCTOR

TABLE OF CONTENTS

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

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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 time.

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.

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PROBABILITYIs the measure of how likely an event is to occur.

FORMULA:

P= EVENTS (specific result of a probability experiment )OUTCOMES ( possibleresult of a probability experiment )

EXAMPLE:

What is the probability of getting head when flipping a coin?

SOLUTION:

P(head )=12

=0.05

=50%

PROBABILITY NUMBER LINE

Impossible unlikely equal chances likely certain

0 0.5 1

0% 50% 100%

2

COMPLEMENTARY PROBABILITY

Is not of that event. The sum of the probability is equal to 1.

FORMULA:

P(A) + P(B) = 1 or P(not A) =1 – P(A)

EXAMPLE:

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.

SOLUTION:

PS(1,4,9,16,25,36,49,64,81,100,121,144)

P(not perfect square)=1 – P(perfect square)

=150150 -12150

= 138150

= 2325 = 0.92 or 92%

3

JOINT PROBABILITY

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)

FORMULAS:

Mutually exclusive eventP(A and B)=0P(A or B)=P(A) + P(B)

Non-mutually exclusive eventP(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 34135

probability of having a type O blood and a 1645 probability of having a type A blood. What

is the probability that a teacher has type A or O type of blood?

Solution:

P(AUB)=P(A) + P(B)

= 34135

+ 1645

= 34135+48135

= 82135 = 0.6074 or 60.74%

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EXAMPLE: Non-mutually exclusive event

What is the probability when you shuffle a deck of cards to get a black card or a 7 card?

Solution:

P(black card or 7 card)=P(BC) + P(7C) – P(BC 7C)Ո

= 2652+452

− 252

= 3052−252

=2852

= 713= 0.5384 or 53.84%

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CONDITIONAL PROBABILITY

Is the probability of an event (A), given that another event (B) has already occurred.

FORMULA:

P(A and B)=P(A) × P(B/A) or P(A/B)=P (B∧A)P(B)

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?

Solution:

P(s/c)=P (CՈS)P(C)

= 35%70%

= 12

= 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)

= 57× 47

= 2049 =0.4081 or 40.81%

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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.

FORMULA:

P=nCr( p)r(q)n−r

Where:

p = prob. of success

q = prob. of failure

n = no. of trials

r = successful outcomes

EXAMPLE:

A die is thrown 6 times. If getting an odd number is a success, what is the probability of 5 successes?

Solution:

p = 36∨12

q = 36∨12

n = 6

r = 5

P= 6C5( 12)5

( 12)6−5

= 332

= 0.0937 or 9.37%

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