Probability.ppt

PROBABILITY Course instructor Saima Sharif Siddiqui

PROBABILITY Meaning of probability: • Chance • Possibility Probability is a numerical measure of the chance that an event “The ratio of the number of favorable cases to the number of all cases .” A denoted by P(A) is defined as the ratio we write P(A) = Number of favourable outcomes Total number of possible outcomes

Properties • 1: The probability ranges between 0 and 1 • 2: The probability of an event that cannot occur is 0. (An event that cannot occur is called an impossible event.) • 3: The probability of an event that must occur is 1. (An event that must occur is called a certain event.)

Definition Sample space: collection of all possible outcomes of an experiment. sample space represented by the symbol S Let a,b,c The sample space of the experiment is (a, a) , (a, b), (a, c) (b, a) , (b, b) , (b, c) (c, a), (c, b) , (c, c) The sample space has S = 9 equally likely outcomes

Definition Event: An event is a subset of a sample space Experiment is the process of obtaining observations or measurements. Outcome: A single result of an experiment is called an outcome

Probability of dice

Sample space for rolling a die once

Find probability of 5 if I roll a die. A die has 6 numbers. There is only one 5 on a die, so P(A) = Number of favorable outcomes Total number of possible outcomes

P(5) = 1/6

Getting an even number if I roll a die Even numbers are 2, 4, 6. So

Total Number of possible outcomes = 6 Number of favorable outcomes = 3 P = No of favourable outcomes No of possible outcomes P(even) = 3/6 = 1/2

Question :1 What is the probability of choosing a vowel from the alphabet? Solution: Total Number of possible outcomes = Number of favorable outcomes =

P(E) = No of favorable outcomes No of possible outcomes P(vowel) =

Question : 2 A number from 1 to 11 is chosen at random. What is the probability of choosing an odd number? Solution: Total Number of possible outcomes = Number of favorable outcomes =

P(E) = No of favorable outcomes No of possible outcomes P(odd) =

Practice Question 3: A glass jar contains 6 red , 5 green , 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?

Practice Question 4: A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on each color?

Possible outcomes for rolling a pair of dice

Example (sample spaces chart) • Find the sample space for rolling two dice. Die1

1 2 3 4 5 6

Die 2 3 4

1

2

5

6

(1,1)

(1,2)

(1,3)

(1,4)

(1,5)

(1,6)

(2,1)

(2,2)

(2,3)

(2,4)

(2,5)

(2,6)

(3,1)

(3,2)

(3,3)

(3,4)

(3,5)

(3,6)

(4,1)

(4,2)

(4,3)

(4,4)

(4,5)

(4,6)

(5,1)

(5,2)

(5,3)

(5,4)

(5,5)

(5,6)

(6,1)

(6,2)

(6,3)

(6,4)

(6,5)

(6,6)

sample spaces for two dice S = {(1,1),(1,2),

Example If two fair dice are thrown ,what is the probability 1.

a double six 2. Sum of 8

Cards

♠

Spades (13 Cards)

♣

Clubs (13 Cards)

♥ Hearts (13 Cards)

♦ Diamonds (13 Cards)

Sample space for the card

The event the king of hearts is selected

1 / 52

The event a king is selected

The event a heart is selected

The event a face card is selected

Rules of Probability  Complement Rule  Addition Rule  Multiplication Rule

Laws of Probability General Addition Rule For any two events A and B P(A  B) = P(A) + P(B) – P(A  B)

Example • If one card is selected at random from a deck of 52 playing card what is probability that the card is diamond or a face card or both?

Diamond or a face card

Practice There are 30 boys and 20 girls in a class. 20 boys and 10 girls use their own transport to reach the college. If a student is selected at random find the probability that the student selected is a boy or a student using his or her own transport to reach the college.

Addition Rules The Addition rule of probability The probability that event A or B will occur when an experiment is performed is given by P(A or B) = P(A) + P(B) OR P(AUB) = P(A) + P(B)

Unions and Intersections

S

A A

B

A

Complementation Rule P(A) = 1- P( A ) Definition: The complement of an event A is the set of all outcomes in the sample space that are not included in the outcomes of event A. The complement of event A is represented by A (read as A bar).

Example 1: A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on a sector that is not red after spinning this spinner? Sample Space: {yellow, blue, green, red} Probability = Number of favourable outcomes Total number of possible outcomes P(R) = 1 /4 P(not red) = 1 – P(R) P (not red) = 1 – 1 /4 P(not red) = 3/4

Practice Q1. A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is not a king?

Practice Q2. A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is not a club?

Practice 3. A glass jar contains 20 red marbles. If a marble is chosen at random from the jar, what is the probability that it is not red? 4. A single 6-sided die is rolled. What is the probability of rolling a number that is not 4?

Multiplication Rule of probability • Independent Events Suppose A and B are two events in an experiment. If A firstly and B secondly. If the probability of second event B has not changed due to first event A , then the events A and B are called independent. The probability of these two independent events P(A and B) = P(A) P(B) P(A∩B) = P(A)P(B)

Example Suppose two cards are drawn one by one, from a pack of 52 cards If the events are A = a king is drawn in first attempt P(A) = 4 / 52 Whereas the first card is replaced be fore drawing the second card. B = a queen is card drawn in second attempt P(B) = 4 / 52 P(A∩B) = P(A)P(B) = (4/52) (4/52)

Practice1: A jar contains 3 red, 5 green, 2 blue and 6 yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and a yellow marble? Solution: Total Number of possible outcomes = Number of favorable outcomes = P(A) = Number of favorable outcomes Total number of possible outcomes P(green) = 5 / 16 P(yellow) = 6 / 16 (PA∩B) =P(A) P(B) P(green and Yellow) = P(Green) P(yellow) = (5 /16) (6 /16) 30 / 256

Practice 2: A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and an eight?

Multiplication Rule • Dependent Events Suppose A and B two events in an experiment. If A firstly and B secondly event and if the probability of second event B changes due to the first event A . Then these events are said to be dependent events. The probability of these two dependent events P(A and B) = P(A) P(B) P(A∩B) = P(A) P(B)

Example Dependent Events Suppose two cards are drawn one by one from a pack of 52 cards. If A = a card drawn in first attempt is red P(A) = 26 / 52 Whereas the first card is not replaced B = a card drawn in second attempt is black P(B) =26 / 51 P(A∩B) = P(A)P(B) = (26 / 52) (26 / 51)

Practice 1 A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack? Let A represent the event that the card selected is a queen P(queen on first pick) = 4 / 52 Let B represent the event that the card selected is a jack P(jack on 2nd pick given queen on 1st pick = 4 / 51 P(A∩B) = P(A) P(B) P(queen and Jack) = (4 / 52) (4 /51) = 16 / 2652

Practice 2 In a shipment of 20 computers, 3 are defective. Three computers are randomly selected and tested. What is the probability that all three are defective if the first and second ones are not replaced after being tested? P(3 defectives) = (3 / 20) (2 / 19) (1 / 18) = 6 / 6840

Practice 3 Four cards are chosen at random from a deck of 52 cards without replacement. What is the probability of choosing a ten, a nine, an eight and a seven in order?

Practice 4 Three cards are chosen at random from a deck of 52 cards without replacement. What is the probability of choosing 3 aces?