PROGRAM DIDIK CEMERLANG AKADEMIK SPM ADDITIONAL MATHEMATICS FORM MODULE PROBABILITY
PROBABILITY.0 Concept Map Contents Page. Probability Of An Event. Probability Of Two Events. 4. Probability of Mutually Exclusive Events.4 Probability Of Independent Events. - SPM Questions Assessment test Answers 0
.0 CONCEPT MAP. Experiment A process to obtain observations The result of an experiment Possible outcomes Sample space, S The set of all possible outcomes An Event, A Mutually Exclusive Events Independent Event PROBABILITY P(A) = P(A B) = P(A) + P(B) P(A B) The complement of the event A. P(A ) = P(two events A and B that are mutually exclusive) is P(A B) = P(A) + P(B) A B= P(two or three independent events) P(A B) = P(A).P(B) P(A B) = P(A).P(B).P(C)
. PROBABILITY OF AN EVENT Example. Question Box A contains black balls, green balls and red balls. A ball is drawn at random from box A. Calculate the probability that the colour of the ball is (a) black not black (c) yellow Answer (a) Let R represent the event that a black ball is drawn. n(r) P(R) =. n(s) 0 P(the ball is not black) =P(R ) = P(R) = = 0 0 (c) P(the ball drawn is yellow) = 0. An event that is impossible to occur because there are no yellow balls in the box Exercises. No. Questions Answers. A box contains cards where each card is marked with an alphabet from the word TAMBAHAN. If a card is chosen at random, calculate the probability that (a) the card with the alphabet B is chosen. a card with vowel is chosen.. There are 4 red marbles and y green marbles in a bag. A marble is drawn at random from the box. Given that the probability of drawing a green is, calculate the value of y.. A bag contain x green marbles, y blue marbles and brown marbles. A marbles is drawn at random and the probability of getting a brown marble is. Write down the equation relating x and y.
. PROBABILITY OF THE TWO EVENTS Example. Question 4 9 The above figure shows six numbered cards. A card is chosen at random. Calculate the probability that the number on the chosen card (a) is a multiple of and a factor of is a multiple of or a factor of. Answer Let A represent the event that the number on the chosen card is a multiple of, and B represent the event that the number on the chosen card is a factor of. A = {,, 9}, n(a)= B = {,, 4, }, n(b) = 4 A B = {, } A B = {,, 4,, 9} (a) P(A B) =. P(A B) = Alternative method P(A B) = P(A) + P(B) P(A B) 4 = =. Exercises No. Questions Answers. A dice is thrown once. Calculate the probability that the score on the dice either an odd number or a prime number.. A card is chosen at random from a bag which contains the different letters of the alphabet. Find the probability that (a) the card chosen has a letter from the word BAHASA the card chosen is not a letter from the word KACA. The set X and Y are given as follows: X = {,,,, 9} Y = {, 4, } A number is chosen at random from set X and another number from set Y. Calculate the probability that the sum of the number is 9 or the product of the number is. 4
. PROBABILITY OF MUTUALLY EXCLUSIVE EVENTS Example. Question A box contains red balls, yellow balls and 4 green balls. A ball is chosen at random from the box. Calculate the probability that the balls drawn neither a yellow nor a green. P (yellow) =. Solution P(green) = 4 P(yellow or green) = + 4 =. Exercises No. Questions Answers. A fair dice is thrown. Let x be the event when the dice shows and Y be the event when the dice shows an even number. (a) Are the two events mutually exclusive? Find the probability that or even number is the outcome.. A marble is drawn at random from a box containing black marbles, 4 green marbles and white marbles. (a) What is the probability of drawing a black or a green marble? What is the probability of drawing neither a black nor a white marble?. Box T contains three card numbered, and. Box U contains three card numbered, and 4. A card is drawn at random from box T and at the same time, another card is drawn from box U. Calculate the probability that the two numbers drawn have the same value or a sum of.
.4 PROBABILITY OF INDEPENDENT EVENTS Example 4 No Questions Solutions. 4 Black Box C contains 4 black marbles and 4 0 0 Black yellow marbles. A marbles is chosen at random from box C, its colour is Yellow 0 noted and the marbles is noted and the 4 marbles is returned to the box. Then a Black 0 second marbles is chosen. Determine 0 Yellow the probability that Yellow (a) (c) both the marbles are black. the two balls are of different colours. at least one of the balls chosen is yellow. 4 4 4 (a) P(black black)= = 0 0 P(same colours) = P(black black) + P(yellow yellow) 4 = + =. 0 0 4 (c) P(both blacks) = = 0. The probability that participants K, L and M will win a dancing contest are, and respectively. If the events of each participant winning are independent, calculate the probability that (a) only L wins, two participants win. (a) K L M Win Win ck Win ck lose k Win Lose ck lose lose Win k Lose P(only L wins) = P(K L M ) = P(K ) P(L) P(M ). = Win ck lose Win ck lose = 4 P( participants win) = P(K L M ) + P(K L M) + P(K L M). = =
Exercises 4 No. Questions Solutions. A bag has green cubes and red cubes. Two cubes are drawn from the bag at random, one after the other, without replacement. Calculate the probability that the green cube and a red cube are drawn.. Hasan competes with John in the finals of a squash competition. The competition will end when a player wins three sets. The probability that Hasan will win any set is. Calculate the probability that (a) the competition ends after only three sets. Hashim is the winner after playing four sets.. Bag B contains red balls and yellow balls. A ball is drawn at random from bag B. The ball is then put into bag D that contains 4 red balls and yellow balls. After that, another ball is drawn at random from bag D. Calculate the probability that the ball drawn from bag B is of the same colour as the ball drawn from bag D.
PAST YEAR QUESTIONS. No. Questions Solutions. A box contains white marbles and k black marbles. If a marble is picked randomly from the box, the probability of getting a black marble is. Find the value of k. SPM 04(No.4 / Paper ).. Table shows the number of coloured cards in a box Colour Number of cards Black Blue 4 Yellow Two cards are drawn at random from the box. Find the probability that both cards are of the same colour. SPM.0(No. 4 / Paper ) ASSESSMENT TEST No. Questions Solutions. There are 4 blue balls and y red balls in the box. A ball is drawn at random from the box. Given that the probability of drawing a red ball is, calculate the value of y.. Two dice, one white and one black, are thrown together. Calculate the probability that the score on the white dice is twice the score on the black dice.
No. Questions Solutions. A box contains 40 marbles. The colours of the marbles are green and red. If a marble is drawn at random from the box, the probability that a green marble is drawn is. Calculate (a) the number of red marbles in the box, the number of red marbles that have to be added to the box such that the probability to drawn a red marble becomes. 4. Bag I contains blue marbles and black marbles while bag II contains blue marbles and 4 black marbles. If a marble is chosen at random from each bag, calculate the probability that (a) both the marbles are black, the marble from bag I is blue and the marble from bag II is black. (c) At least one of the marbles chosen is black.. Two six-faced unbiased dice are thrown together. Calculate the probability that (a) the sum of two numbers is. The difference of two numbers is, (c) The sum of two numbers is or The difference of two numbers is.. In a soccer match between team B and team D, the result can be a draw or a win for team B or a win for team D. The probability that team B and team D will win are and. In two matches, calculates the probability that team B wins once and draw once. 9
ANSWERS Exercises. (a).. x + y = 0. 44 PAST YEAR QUESTION Exercises.. (a). 4 Exercise. (a) Yes. (a).. 9 9. ASSESSMENT TEST. 4.. 4 4. (a). (a). (c) 9 Exercise 4. 4. (a). 0