![SIMPLE EVENT
A simple event consists of a single outcome that cannot be
further broken down into smaller events. Tossing a single
coin will produce either head or tail [H,T].
COMPOUND EVENT
A compound event is any event combining two or more
simple events. The event that at least one head appears in
tossing a coin twice is a compound event.](https://image.slidesharecdn.com/grade10math1-240428150801-3ba346a5/85/GRADE-10-MATH-Probability-and-Statistics-14-320.jpg)
GRADE 10 MATH Probability and Statistics
- 1. Grade 10 Mathematics Prepared by: Kenth Richard M. Romulo
- 10. The chance that something will happen - how likely it is that some event will happen.
- 11. SAMPLE SPACE A sample space is the set of all possible outcomes of an experiment OUTCOME Each element of a sample space is called an outcome or a sample point. EVENT Any subset of a sample space is an event(E).
- 12. EXPERIMENT: In rolling a die. A die has faces numbered 1 to 6. Hence, S = {1,2,3,4,5,6}
- 13. EXPERIMENT: In rolling a die. A die has faces numbered 1 to 6. Hence, S = {1,2,3,4,5,6} Event (E): getting an even number E = {2, 4, 6}
- 14. SIMPLE EVENT A simple event consists of a single outcome that cannot be further broken down into smaller events. Tossing a single coin will produce either head or tail [H,T]. COMPOUND EVENT A compound event is any event combining two or more simple events. The event that at least one head appears in tossing a coin twice is a compound event.
- 15. P (E) = 𝑛 ( 𝐸 ) 𝑛 ( 𝑆 ) Where, E = event S = Sample Space n ( E ) = Number of Events n ( S ) = Total Number or values in the Sample Space
- 16. Experiment: throwing a die Event (E): getting an even number Solution: A die has faces numbered 1 to 6. Hence, S = {1,2,3,4,5,6} E = {2, 4, 6} P (E) = 𝑛 ( 𝐸 ) 𝑛 ( 𝑆 ) P (E) = 3 6 P (E) = 1 2
- 17. Experiment: Tossing a coin Event (E): getting a head Solution: A coin has two faces. S = {H, T} n(S) = 2 E = {H} n(E) = 1 P (E) = 𝑛 ( 𝐸 ) 𝑛 ( 𝑆 ) P (E) = 1 2
- 18. AND PROBABILITY And means that the outcome has to satisfy both conditions at the same time. OR PROBABILITY Or means that the outcome has to satisfy one condition, or the other condition, or both at the same time.
- 19. Find the probability of “getting a 6 and a 1” when two dice are rolled is an event consisting of (1, 6), (6, 1) as outcomes.
- 20. The first die falls in 6 different ways and the second die also falls in 6 different ways. Thus, using the fundamental counting principle, the number of outcomes in the sample space is 6x6 or 36. The outcomes in the sample space are: {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3),…,(6, 5), (6, 6)}. Take note that “getting a 6 and a 1” when two dice are rolled is a compound event consisting of {(1, 6), (6, 1)} as outcomes.
- 21. n (S) = 36 n (E) = (6,1), (1, 6) P (E) = 𝑛 ( 𝐸 ) 𝑛 ( 𝑆 ) P (E) = 2 36 P (E) = 1 18
- 23. Find the probability if two dice are tossed, what is the probability that the sum will be 4 or 5? Solution: Let E1 be the event that the sum of 4 will come out and E2 be the sum of 5 will come out. The number of elements in the sample space S, n(S), is equal to 6x6=36 or n(S) = 36.
- 24. In this problem, we have the following outcomes for each event: E1 = {(2,2), (1,3), (3,1)} E2 = {(2,3), (3,2),(1,4),(4,1)} n(E) = 3+4 = 7 P (E) = 𝑛 ( 𝐸 ) 𝑛 ( 𝑆 ) P (E) = 7 36
- 25. Let E1 and E2 be the two successive events in a sample space S. If the outcome of E2 is affected by the prior occurrence of E1, then E2 and E1 are dependent events. However, if the outcome of E2 is not affected by the prior occurrence of E1, then E1 and E2are independent events. We use the symbol P (E2 / E1) as notation to read as, “the probability of the occurrence of E2 on the condition that E1 has already occurred.
- 26. A man draws a card twice from an ordinary deck wherein the first event (E1) is getting a red card and getting black king is the second event (E2) with the following conditions: A. The first card was returned before the second card is drawn. B. The first card was not returned.
- 27. A. Since the first card was replaced then the events are independent. This means the event of getting a black king is not affected by the prior occurrence of the event of drawing a red card. B. Since there was no replacement after the first event E1 has been performed, then second event E2 is affected and is dependent to E1.
- 28. Since events are sets, can be combined to form new events by using the set operation of union. It can be illustrated by means of Venn diagram.
- 29. A die is tossed. Sample Space (S) = {1, 2, 3, 4, 5, 6} Let A = event that an odd number occurs B = event that a number greater than 4 occurs Determine the elements of A and B. then, find A U B and draw a Venn diagram to illustrate it. Solution: A = {1, 3, 5} B = {5, 6} A U B = {1, 3, 5, 6}
- 30. Two events A and B intersect if there are elements common to both A and B. It is denoted by A ∩ B.
- 31. A fair die is rolled. The sample is S = {1, 2, 3, 4, 5, 6} Let A = event “even number turns up” B = events “the number that turns up is greater than 2” Find A n B and draw a Venn diagram to illustrate it.
- 32. A = {2, 4, 6} B = {3, 4, 5, 6} Thus, A ∩ B = {4, 6}
- 33. The cardinality of a set refers to the number of elements of the set. The cardinality of set A is denoted by |A|.
- 34. Let A = {2, 4, 6, 8, 10}. Find |A|? Solution: Since there are 5 elements in set A, therefore, |A| = 5.
- 35. The sample space S consist of the six possible outcomes of die tossed once. S = {2, 3, 4, 5, 6, 7}. Find |S|? Solution: Sample space S contains 6 outcomes, therefore, |S| = 6.
- 36. Directions: Choose the letter of the correct answer. 1.What is a subset of a sample space called? a) Element b) Event c) Set d) Cardinality 2. How is union of sets A and B denoted? a) AB b)A U B c)A d)A+B
- 37. 3. The union and intersection can be illustrated by means of ____________. a) Fish bone b)Venn Diagram c)Flow Diagram d)Organizational Chart 4. What is described if there are elements common to both events A and B? a)Complement b)Union c)Cardinality d)Intersection
- 38. 5. What is the probability that an event can never be happened? a) 0 b)between 0 and 1 c)1 d)cannot be determined 6. Which type of events is affected by prior occurrence of the other event? a)Dependent b)Mutually exclusive c)Independent d)Simple
- 39. 7. Which condition affects the drawing of cards twice from an ordinary deck? The ___ a) first card was not returned b)first card was returned c)second card is returned d)Second card is drawn 8. Which is NOT true about compound event? a)The event that at least one head appears in tossing a coin twice. b)Then probability of “getting a 6 and a 1” when two dice are rolled. c)It is a combination of two simple events. d)It is the intersection of two events.
- 40. 9. What is described when an event contains exactly one sample point or outcome? a) Simple event b)Independent event c)Compound event d)Mutually exclusive 10. What term describes the manner of drawing cards, tossing, or rolling of a coin and a die repeatedly? a)Outcome b)Sample point c)Experiment d)Event
- 41. 11. Which shade illustrates the union of three events A, B and C in a sample space?
- 42. 12. What is the probability of getting 2 or 6 in rolling a die? a) 5/6 b) 1/6 c) 2/3 d) 2/6 or 1/3 13. What is the intersection of events A and B if A are whole numbers up to 10 and B are numbers divisible by 3? a) A ∩ B = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } b) A ∩ B = { 0, 2, 4, 6, 8, 10 } c) A ∩ B = { 0, 1, 3, 5, 7, 9 } d) A ∩ B = { 3, 6, 9 }
- 43. 14. Picking a king of hearts in a deck of cards a) Simple event b) Compound event 15. Which of the following Venn diagrams illustrates the intersection of events A and C?