Mr. Mark Anthony Garcia, M.S. Mathematics Department De La Salle University

Probability Probability is the likelihood that a certain event will happen. The probability value may be written in relative frequency form (A value between 0 and 1) or in percentage form (relative frequency times 100) or in fraction form (numerator is less than or equal to denominator)

Null Event A null event or null space is an event that is impossible to happen. The null event is denoted by the symbol Ø.

Example 1: Null Event  The event of obtaining the number 7 when a die is thrown.  The event of obtaining a heart card which is black in a deck of playing cards.  The event that there will be no January 1 in a year.

Probability: Properties  The probability of an event A, denoted by, P(A), is always between 0 and 1, inclusive.  The probability of the null event ø, denoted by, P(ø) is 0.  The probability of the sample space S, denoted by, P(S) is 1.

Priori Probability

Example 2: Priori Probability

Posteriori Probability

Example 3: Posteriori Probability  Consider the experiment of tossing a coin.  Let N = 20 be the number of tosses.  Let E be the event of obtaining a head. Then E = {H}.  Consider the results table in the next slide.

Example 3: Posteriori Probability 1. H6. T11. T16. H 2. H7. H12. T17. T 3. T8. H13. H18. H 4. T9. T14. T19. T 5. H10. H15. H20. H

Example 3: Posteriori Probability No. of times H occurred in first N trials First N trialsPosteriori Probability

Probability Formula 1: Basic Rule

Example 4: Probability Formula 1  A coin is tossed twice. What is the probability that at least 1 head occurs?  The sample space is S = {HH, HT, TH, TT}.  The event A is the event that at least one head occurs in the two tosses of the coin.

Example 5: Probability Formula 1  A die is rolled once, what is the probability that an even number occurs?  The sample space is S = {1,2,3,4,5,6}.  The event A is the event that an even number occurs in the roll.

Example 6: Probability Formula 1  If a card is drawn from an ordinary deck, find the probability that a heart card is drawn.  The sample space is the set of 52 playing cards.  The event A is the event that the card drawn is a heart card.

Example 7: Probability Formula 1

Set Operations

Example: Set Operations

Mutually Exclusive Events

Example 8: Mutually Exclusive Events

Probability Formula 2: Additive Rule

Example 9: Probability Formula 2 The probability that a student passes Algebra is 2/3, and the probability that he passes English is 4/9. If the probability of passing at least 1 subject is 4/5, what is the probability that he will pass both subjects?

Example 10: Probability Formula 2 What is the probability of getting a total of 7 or 11 when a pair of dice is tossed?

Probability Formula 3: Complementary Rule

Example 11: Probability with Combinations If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary, what is the probability that A. the dictionary is selected? B. 2 novels and 1 book of poems are selected?

Example 12: Probability with Permutations If a permutation of the word “white” is selected at random, find the probability that the permutation A. Begins with consonant; B. Ends with a vowel; C. Has the consonants and vowels alternating.

Example 13: Probability in Two- way tables A random sample of 200 adults are classified below according to sex and level of education attained. MaleFemaleTotal Elementary Secondary College Total

Example 13: Probability in Two- way tables If an adult is selected at random, what is the probability that A. the adult is a male or has reached elementary level only B. the adult is a female a and reached up to secondary level C. the adult did not reach college level

Probability Formula 4: Conditional Probability

Example 14: Probability Formula 4 Suppose our sample space S is the set of 4 th year high school student in a small town who took the UPCAT college entrance examination. Let us categorize them according to sex and whether they passed or not.

Example: Probability Formula 4 PassedFailedTotal Male Female Total

Example 14: Probability Formula 4

Probability Formula 5: Multiplicative Rule

Example 15: Probability Formula 5 Suppose that we have a fuse box containing 20 fuses, of which 5 are defective. If 2 fuses are selected at random and removed from the box in succession without replacing the first, what is the probability that both fuses are defective?

Independent Events Two events A and B are independent if the occurrence of event A does not affect the occurrence of event B.

Example 16: Independent Events Consider the experiment of tossing a coin twice. The first toss of the coin (event A) does not affect the outcomes of the second toss (event B). Thus, A and B are independent events.

Example 16: Independent Events Consider the departure (event A) and arrival (event B) of airplanes. If the plane departs on time, then it will probably arrive on time. Therefore, A and B are dependent events.

Probability Formula 6: Independent Events

Example 17: Probability Formula 6 A small town has one fire engine and one ambulance available for emergencies. The probability that the fire engine is available when needed is 0.98, and the probability that the ambulance is available when called is Find the probability that both the ambulance and the fire engine will be available.