Secondary 3 Mathematics Advanced Exercise 06: Introduction to Probability

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ADVANCED EXERCISE 06: INTRODUCTION TO PROBABILITY

1. Two red “Aces” are taken away from a deck of 52 playing cards. One card is then drawn. Find the following probability.

(a) A number card is drawn. (b) A diamond number card is drawn. (c) A diamond card or a number card is drawn. (d) A card which is neither a black “Jack” nor an “Ace” is drawn. 2. A two-digit odd number from 11 to 99 is chosen randomly. (a) How many possible odd numbers are there from 11 to 99 inclusive? (b) Find the probability that the number chosen is greater than 90. (c) Find the probability that the number chosen is divisible by 3. (d) Find the probability that the number chosen is relatively prime with 6. (Two numbers ,a b are relatively prime if they have no common factor other than 1.)

3. The first day of August in a certain year is Monday. One day in August is chosen at random. (a) Find the probability that the day chosen is Monday. (b) Find the probability that the day chosen is Sunday and after 10th August.

Secondary 3 Mathematics Advanced Exercise 06: Introduction to Probability

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4. In a room, there are 3 men and some women. When 3 more women enter the room, the probability of choosing a man

decreases by 1

12.

(a) Find the original number of women in the room. (b) Now the 3 extra women have entered the room. If 1 person is chosen at random, find the probability that a man

would be chosen. 5. A card is taken out from a bag containing 5 cards, each with a different letter written on it. It is given that the

probabilities of getting a “b”, “c” and “d” are all 0.2, while that of getting a vowel is 0.4. (a) Write down one possible set of alphabet shown on the cards. (b) Find the probability that either a letter “b” or a letter “f” is drawn. 6. The following table shows the distribution of the numbers got when throwing an identical dice 600 times.

Number 1 2 3 4 5 6 Frequency 100 100 105 90 110 95

Find the theoretical and experimental probability of getting a number greater than 4.

Secondary 3 Mathematics Advanced Exercise 06: Introduction to Probability

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7. One fair dice is thrown. (a) (i) Find the probability of getting a prime number. (ii) Given that the number shown up is odd, find the probability of getting a prime number. (b) Two fair dice are thrown. Find the probability of the following events.

(i) The sum of the two dice is greater than 9. (ii) The product of the two dice is odd. (iii) The difference of the two numbers is less than 2. (iv) At least one prime number shows up.

(c) Three dice are thrown. Find the probability that all numbers are the same. 8. There are 3 red balls, 2 orange balls and 1 white ball in the bag. Two balls are taken out randomly with replacement. (a) Show all possible outcomes by a table. (b) Find the probability of the following events.

(i) Neither ball is red. (ii) Only one of the balls is orange. (iii) The first ball is orange.

Secondary 3 Mathematics Advanced Exercise 06: Introduction to Probability

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9. Jacky and Jenny play the game “hand, stone and scissors”. Find the theoretical probability that Jacky wins the game (Note that draw is also a possible outcome.)

10. Peter is guessing the answer for two True-False questions and 1 multiple-choice question with 4 choices. Fir each

question, only one answer is correct. (a) Find the following probabilities. (i) He answers the three questions correctly. (ii) He answers exactly one question correctly. (b) Given that he answers the multiple-choice questions correctly. Find the probability that he will get

(i) one correct answer in total. (ii) two correct answers in total.

11. The details of a game are as follows: Three fair coins are tossed. If all coins show the same face, one will get $10.

Otherwise one will get $2 (a) Find the probability that all coins show the same face. Hence find the expected amount of money got. (b) If one needs to pay $x for the game, find the range of values of x such that the game is favorable to the player.

Secondary 3 Mathematics Advanced Exercise 06: Introduction to Probability

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12. Refer to the figure.

(a) Find the ratio of the areas of the two parts. (b) 2 points and 1 point are awarded when the dart hit the inner and outer area respectively. Find the expected

number of points obtained for hitting three darts. (Assume the dart will not hit outside the figure.) 13. Three coins are tossed. (a) Represent all possible outcomes by a tree diagram. (b) Find the probability that the number of heads got is greater than that of tails. (c) Given that one of the coins shows head, find the probability that the other two also show head. 14. 2 students are chosen from a class of 6 boys and 4 girls randomly. (a) Find the probability of (i) choosing two boys. (ii) not choosing two girls. (b) Wilson is one of the boys in the class.

(i) Given that a boy, not Wilson, is chosen from the class, find the probability that Wilson is chosen next. (ii) Given that a girl is chosen, find the probability that Wilson is chosen next.

(iii) Find the probability that Wilson is chosen.

Secondary 3 Mathematics Advanced Exercise 06: Introduction to Probability

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15. Peter throws a coin, with diameter 2 cm, to a board containing squares grids of sides 10 cm as shown in the figure below. Find the probability that the coin will not touch or intersect with the boundary of the grids. (Assume that the board is infinitely long.)

16. There are 2 boxes X and Y. In box X, there are 3 gold coins and 1 silver coins. In box Y, there are 2 gold coins and 2

silver coins. A box is randomly chosen and a coin is randomly taken out. (a) Draw a tree diagram to show all possible outcomes. (b) Find the probability that a gold coin is taken out from box X. (c) Find the probability that a silver coin is taken out. (d) If the process is repeated for 100 times, estimate the number of gold coins taken out.

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17. There are 5 numbers in a bag: 1, 2, 3, 5 and 8. (a) One number is drawn. Find the expected value of the number drawn. (b) Two numbers are chosen without replacement.

(i) Use a table to represent all possible outcomes. (ii) Find the probability of getting a sum less than 6.

(iii) Find the probability of getting at least 1 prime number. (c) Three numbers are drawn. Find the probability that the product of the numbers drawn is even. (d) Four numbers are drawn. Find the probability that the 2 even numbers are taken.

18. Ten people, including student A and B, sit on a round table randomly. (a) Find the probability that student A sits next to student B. (b) Find the probability that student A sits next to, and on the right hand side of student B.

Secondary 3 Mathematics Advanced Exercise 06: Introduction to Probability

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19. n fair coins are tossed. (a) How many different possible outcomes are there? Express your answer in terms of n. (Outcomes with different

order are considered as different outcomes.) (b) If 6n= , find the probability of the following events.

(i) All coins show tails. (ii) At least one head is obtained.

(c) If 7n= , find the probability of the following events. (i) The number of heads obtained is greater than that of tails. (ii) 1 head and 6 tails are got.

20. Three babies are born. Assume equal probabilities of getting a girl or a boy, find the expected number of girls born. 21. Two fair tetrahedron dice with numbers 2, 3, 4 and 5 are thrown. Find the expected number of the following. (a) The sum of the two numbers shown on the 2 dice. (b) The number of “2” got.

Secondary 3 Mathematics Advanced Exercise 06: Introduction to Probability

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22. In an archery game, an arrow is shot onto a board shown in the figure. If the arrow falls onto the grey region, which consists of two squares, a person can get $40. Otherwise he will get a small present which costs $1. The probability

that the arrow falls onto the grey region is 1

4. (Assume all arrows can go to the board.)

(a) Find the length of one side of the square. (b) If one needs to pay $15 for one game, find the expected return. (c) If Timmy has $300, estimate the maximum number of games he can play, assuming that he shots the arrows

randomly.

23. Mr. Chiu has two $0.5 coins, two $1 coins and three $2 coins inside his pocket. He randomly takes 2 coins to pay the

minibus fare which is $2.6. (a) Find the probability that he needs to take more coins in his pocket. (b) Given that he can pay for the fare using the two coins taken out, find the probability that his loss is the smallest. (c) Find the expected amount of money taken out.