$\def\frac{\dfrac}$

The probabilities of students $A, B, C$ to pass an examination are $\frac{3}{4} ; \frac{4}{5}$ and $\frac{5}{6}$ respectively. Find the probability that at least one of them will pass the examination. $\quad$. $\quad(5$ marks $)$

The probability of an event $A$ happening is $\frac{2}{3}$ and the probability that an event $B$ happening is $\frac{3}{8}$ : Given that $A$ and $B$ are independent, calculate the probability thai neither event happens and just one of the two events happens.

Three tennis players $A, B, C$ play each other only once. The probability that $A$ will beat $B$ is $\frac{2}{7}$, that $B$ will beat $C$ is $\frac{1}{3}$ and that $C$ will beat $A$ is $\frac{2}{5}$. Calculate the probability that $A$ wins both games.

A die is rolled 360 times. Find the expected frequency of a factor of 6 and the expected frequency of a prime number. If all the scores obtained in these 360 trials are added together, what is the expected total score?

Draw a tree diagram to list all possible outcomes for a family which has three children. Find the probability that (i) only the first child is a boy (ii) the last child is a boy (iii) the last two children born are boys.

Q11(b) Solution

$\mathrm{X}$ and $\mathrm{Y}$ are two independent events. The probability that the event $\mathrm{X}$ will occur is twice the probability that the event $\mathrm{Y}$ will occur and the probability that $\mathrm{Y}$ will not occur is four times the probability that $\mathrm{X}$ will not occur. Then find the probability that both $X$ and $Y$ will not occur. $\quad(5$ marks)

How many 3 digit numbers less than 400 can you form by using $1,2,3$ and 4 without repeating any digit? If one of these numbers is chosen at random, find the probability that it is divisible by 3 but not divisible by 4 . Find also the probability that a number which is not divisible by 3.

Click for Solution

How many 3 -digit numerals can you form from $3,0,1$ and 6 without repeating any digit? Find the probability of an even number and find the probability that a numeral which is divisible $3 .$

A die is thrown. If the probability of getting a number not less than $\mathrm{x}$ is $\frac{2}{3}$, find $\mathrm{x}$. (3 marks) Click for Solution

Construct the table of outcomes for rolling two die. Find the probability of an outcome in which the score on the first die is less than that on the second die. Find also the probability that the score on first die is prime and the score on the second is even. ( 5 marks) Click for Solution

A die is thrown. If the probability of getting a number less than $\mathrm{x}$ is $\frac{2}{3}$, find $\mathrm{x}$. ( 3 marks) Click for Solution 3(b)

Construct the table of outcomes for rolling two dice, a blue die and a black die. Find. the probability that the score on the blue die is less than that on the black die. Find also the probability that the score on the blue die is prime and the score on the black die is' even. (5 marks)Click for Solution 10(b)

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**Group (2015-2019)**

**1. (2015/Myanmar /q11b )**

The probabilities of students $A, B, C$ to pass an examination are $\frac{3}{4} ; \frac{4}{5}$ and $\frac{5}{6}$ respectively. Find the probability that at least one of them will pass the examination. $\quad$. $\quad(5$ marks $)$

**2. (2015/FC /q11b )**

The probability of an event $A$ happening is $\frac{2}{3}$ and the probability that an event $B$ happening is $\frac{3}{8}$ : Given that $A$ and $B$ are independent, calculate the probability thai neither event happens and just one of the two events happens.

**3. (2016/Myanmar /q11b )**

Three tennis players $A, B, C$ play each other only once. The probability that $A$ will beat $B$ is $\frac{2}{7}$, that $B$ will beat $C$ is $\frac{1}{3}$ and that $C$ will beat $A$ is $\frac{2}{5}$. Calculate the probability that $A$ wins both games.

**4. (2016/FC /q11b )**

A die is rolled 360 times. Find the expected frequency of a factor of 6 and the expected frequency of a prime number. If all the scores obtained in these 360 trials are added together, what is the expected total score?

**5. (2017/Myanmar /q11b )**

Draw a tree diagram to list all possible outcomes for a family which has three children. Find the probability that (i) only the first child is a boy (ii) the last child is a boy (iii) the last two children born are boys.

Q11(b) Solution

**6. (2017/FC /q11b )**

$\mathrm{X}$ and $\mathrm{Y}$ are two independent events. The probability that the event $\mathrm{X}$ will occur is twice the probability that the event $\mathrm{Y}$ will occur and the probability that $\mathrm{Y}$ will not occur is four times the probability that $\mathrm{X}$ will not occur. Then find the probability that both $X$ and $Y$ will not occur. $\quad(5$ marks)

**7. (2018/Myanmar /q11b )**

How many 3 digit numbers less than 400 can you form by using $1,2,3$ and 4 without repeating any digit? If one of these numbers is chosen at random, find the probability that it is divisible by 3 but not divisible by 4 . Find also the probability that a number which is not divisible by 3.

Click for Solution

**8. (2018/FC /q11b )**

How many 3 -digit numerals can you form from $3,0,1$ and 6 without repeating any digit? Find the probability of an even number and find the probability that a numeral which is divisible $3 .$

**9. (2019/Myanmar /q3b )**

A die is thrown. If the probability of getting a number not less than $\mathrm{x}$ is $\frac{2}{3}$, find $\mathrm{x}$. (3 marks) Click for Solution

**10. (2019/Myanmar /q10b )**

Construct the table of outcomes for rolling two die. Find the probability of an outcome in which the score on the first die is less than that on the second die. Find also the probability that the score on first die is prime and the score on the second is even. ( 5 marks) Click for Solution

**11. (2019/FC /q3b )**

A die is thrown. If the probability of getting a number less than $\mathrm{x}$ is $\frac{2}{3}$, find $\mathrm{x}$. ( 3 marks) Click for Solution 3(b)

**12. (2019/FC /q10b )**

Construct the table of outcomes for rolling two dice, a blue die and a black die. Find. the probability that the score on the blue die is less than that on the black die. Find also the probability that the score on the blue die is prime and the score on the black die is' even. (5 marks)Click for Solution 10(b)

**Answer (2015-2019)**

1. $\frac{119}{120}$

2. $\frac{5}{24},\frac{13}{24}$

3. $\frac{6}{35}$

4. $240,180,1260$

5. (i) $\frac{1}{8}$ (ii) $\frac{1}{2}$ (iii) $\frac{1}{4}$

6. $\dfrac{4}{19}$

7. $\frac{7}{18}, \frac{4}{9}$

8. $\frac{5}{9}, \frac{2}{9}$

9. $2<x\le 3$

10. $\dfrac{5}{12},\dfrac 14$

11. $4<x\le 5$

12. $\dfrac{5}{12},\dfrac 14$

**Group (2014)**

1. How many 3 digit numerals can you form from $1,0,5$ and 6, without repeating any digit? Draw a tree diagram to determine the set of all possible outcomes and find the probability of a numeral which is divisible by $5 .$ (5 marks)

2. A blue die and a black die are rolled. Construct a table of possible outcomes. Find the probability that the total scores a multiple of 3 and find also the probability that the product of the scores on the two dice is divisible by $4 .$ (5 marks)

3. Two dice are rolled. Find the probability of an outcome in which the score on the second die is greater than that on the first. Find also probability that the total score on the two dice is prime. (5 marks)

4. Each of the numbers 1, 2, 3 is painted on a separated ball. The three balls are replaces in a bag and it is shaken to mix of the balls. A ball is drawn and then replaced, after which a second ball is drawn. Find the probability that the first ball has an even number and second ball has an odd number. Find also the probabilities that the first ball has a number less than 3 and the second ball has a number greater than 2, and the sum of the numbers on both balls will be 4. (5 marks)

5. A box contains six cards numbered as 1, 2, 3, 4, 5 and 6. A card is chosen and the card is not replaced. Then another card is chosen. Write down the set of all possible outcomes. Find the probability of getting two numbers where they are both odd numbers. Find also the probability of getting two numbers where the product is an odd number. (5 marks)

6. A bag A contains six red balls and five blue balls. A bag B contains one red ball and two blue balls. A ball is randomly taken from bag A and is then placed in bag B. A ball is then randomly selected from bag B. Find the probability that the first ball is blue and the second ball is red. (5 marks)

7. A box contains 15 balls of which 4 are white, 5 are red and 6 are blue. Two balls are to be drawn at random, in succession, each being replaced after its colour has been noted. Find the probability that both the two balls drawn out are of the same colour. Find also the probability that at least one of the two balls drawn out is not white. (5 marks)

8. There are three boxes A, B and C. A contains 3 white and 1 black balls, B contains 2 white and 2 black balls and C contains 1 white and 3 black balls. From each of the three boxes, one ball is drawn at random. Find the probability that 2 white balls and 1 black ball will be drawn. (5 marks)

9. Out of the 20 appplicants for a job there are 8 women and 12 men. It is desired to selected 3 persons for this job. Find the probability that at least one person of the selected persons will be a men. (5 marks)

10. In a car park, there are 7 white cars, 5 black cars and $x$ red cars. One car is chosen at random. Given that the probability that it will be red is $\frac{1}{5}$. Calculate the value of $x .$ Using your value of $x$; find the probability that the first three cars that will leave the car park will be the different color. Find also the be the probability that the first three cars that will leave the car park will be the same color. (5 marks)

11. The probabilities of three teams, $A, B$ and $C$, winning a football competition are $\frac{1}{4}, \frac{1}{8}$ and $\frac{1}{10}$ respectively. Assuming only one team can win, find the probability that either $A$ or $B$ wins. Find also the probability that neigher $A$ nor C wins. (5 marks)

**Answer (2014)**

1. $\dfrac{5}{9}$

2. $\dfrac{1}{3}$,$\dfrac{5}{12}$

3. $\dfrac{5}{12}$,$\dfrac{5}{12}$

4. $\dfrac{2}{9}$,$\dfrac{2}{9}$,$\dfrac{1}{9}$

5. $\dfrac{1}{5}$,$\dfrac{1}{5}$

6. $\dfrac{5}{44}$

7. $\dfrac{77}{225}$,$\dfrac{209}{225}$

8. $\dfrac{13}{32}$

9. $\dfrac{271}{285}$

10. $x=3,\dfrac{3}{26}$,$\dfrac{46}{455}$

11. $\dfrac{3}{8}$,$\dfrac{13}{20}$

**Group (2013)**

1. Draw a tree diagram to list all possible two-digit numerals which can be formed by using the digits $2,3,5$ and 6 without repetition. If one of these numerals is chosen at random, find the probability that it is divisible by $13 .$ Find also the probability that it is either a prime number or a perfect square. (5 marks)

2. Box $A$ contains 4 pieces of paper numbered as $1,2,3$ and 4. Box $B$ contains 3 pieces of paper numbered as 5,6 and 7. One piece of paper is chosen at random from box $A$ and then one piece of paper is chosen at random from box $B$. Draw a tree diagram to list all possible outcomes of the experiment. Find the probability that the product of the two numbers chosen is at most 10 . Find also the probability that the sum of the two chosen numbers is greater than their product. (5 marks)

3. Make a table of order pairs (first die, second die) for rolling two dice. Use the table to find the probability that the sum of the scores is odd, the product of the scores is greater than 15 and the product of the score is a multiple of $6 .$ (5 marks)

4. Two fair dice are thrown. Calculate the probability that the sum of score is even, the product of the scores is greater than 20 and the product of the scores is a multiple of 6. (5 marks)

5. The probabilities of students $A, B$ and $C$ to pass an examination are $\frac{2}{3}, \frac{3}{5}$ and $\frac{1}{2}$ respectively. Find the probability that only $A$ and $C$ will pass the examination and that at least one of them will pass the examination. (5 marks)

6. Three players $A, B, C$ play each other only once. The probability that $A$ will win $B$ is $\frac{1}{3}$ that $B$ will win $C$ is $\frac{2}{5}$ and that $C$ will win $A$ is $\frac{2}{7}$, Calculate the probability that $A$ or $B$ wins both games. (5 marks)

7. A box contains 2 black, 4 white and 3 red balls. One ball is drawn at random from the box, and kept aside. From the remaining balls in the box, another ball is drawn at random and kept besides the first. This process is repeated till all the balls are drawn from the box. Find the probability that the balls drawń, are in the sequence of 2 black, 4 white and 3 rẹd. (5 marks)

8. A box contains 12 discs of which 3 are white, 4 are red and 5 are blue. Two discs are to be drawn at random, in succession, each being replaced after its colour has been noted. Find the probability that both the two discs drawn out are blue. Find also the probability that exactly one of the two discs drawn out is blue. (5 marks)

9. Three groups of children contain respectively 3 girls and 1 boy, 2 girls and 2 boys, 1 girl and 3 boys. One child is selected at random from each group. Find the probability that three selected are all boys. (5 marks)

10. Three groups of children contain respectively 3 girls and 2 boys, 3 girls and 3 boys, 3 girls and 4 boys. One child is selected at random from each group. Find the probability that three selected are all girls. (5 marks)

11. Three groups of people consist of 3 women and 2 men, 3 women and 3 men, and 3 women and 24 men respectively. If one person is selected at random from each group, find the probability that three people selected are all women. (5 marks)

**Answer (2013)**

1. $\frac{1}{4}, \frac{1}{3}$

2. $\frac{1}{3}, \frac{1}{4}$

3. $\frac{1}{2},\frac{11}{36},\frac{5}{12}$

4. $\frac{1}{2}, \frac{1}{6}, \frac{5}{12}$

5. $\frac{2}{15}, \frac{14}{15}$

6. $\frac{53}{105}$

7. $\frac{1}{1260}$

8. $\frac{70}{144}$

9. $\frac{3}{32}$

10. $\frac{9}{70} \quad$

11. $\frac{3}{90}$

#### ** Group (2012)**

$\quad$ | $\,$ | |
---|---|---|

1. | How many two digits numerals can you form from $1,2,3,4$ and 5 . Find the probability of a numeral which is divisible by 4 and also find the probability of a numeral which is a prime number. (5 marks) | |

2. | Draw a tree diagram to list all possible two-digit numerals which can be formed by using the digits $2,3,5$ and 6 without repeating any digit. If one of these numerals is chosen at random, find the probability that it is divisible by $13 .$ Find also the probability that it is either a prime number or a perfect square. (5 marks) | |

3. | A cion is tossed four times. Head or tail is recorded each time. Draw a tree diagram. Find the probabilities of exactly one tail and at least one tail. (5 marks) | |

4. | Construct the table of outcomes for rolling two dice. Find the probability that the score on the second die is greater than that on the first. Find also the probability that the score on one die is prime and the score on the other is even. (5 marks) | |

5. | A spinner is equally likely to point to any one of $1,2,3,4 .$ Make a table of ordered pairs (First spin, Second spin). Find $P$ (second score is greater than 2 ) and $P$ (total score not less than 5 ). (5 marks) | |

6. | A spinner is equally likely to point to any one of the numbers $2,3,4$ and 5 . The spinner is spun once and then a die is rolled. Make a table of ordered pairs (Spinner, Die). Find the probability that the sum of two numbers is prime and that the product of two numbers is a multiple of 2 but not multiple of 3. (5 marks) | |

7. | A bag contains 15 balls of which 4 are white, 5 are green and 6 are blue. Two balls are to be drawn at random, in succession, each being replaced after its colour has been noted. Calculate the probability that the two balls will be of the same colour. (5 marks) | |

8. | A bag contains 25 balls of which 12 are white, 5 are red and 8 are blue. Three balls are drawn at random, but not replaced. Calculate the probability that all three balls will be different colour. (5 marks) | |

9. | Eleven cards, bearing the letters $E, X, A, M, I, N, A, T, I, O, N$ are placed in a box. Three cards are drawn out at random without replacement. Calculate the probabilities that the three cards bear the letters $A, I, M$ in that order and in any order. (5 marks) | |

10. | Out of 13 applicants for a job there are 5 women and 8 men. It is desired to select 2 persons for the job. Find the probability that at least one of the selected persons will be a women. (5 marks) | |

11. | A spinner is equally likely to point to any one of the numbers $1,2,3,4,5,6,7,8,9$, 10. What is the probability of scoring a number divisible by $2 ?$ If the arrow is spum 1000 times, how many would you expect scoring a number not divisible by $2 ?$ (5 marks) |

#### ** Answer (2012)**

$\quad$ | $\,$ | |
---|---|---|

1. | $\frac{1}{5}, \frac{7}{25}$ | |

2. | $\frac{1}{4}, \frac{1}{3}$ | |

3. | $\frac{1}{4}, \frac{15}{16}$ | |

4. | $\frac{5}{12}, \frac{7}{36}$ | |

5. | $\frac{1}{2}, \frac{5}{8}$ | |

6. | $ \frac{3}{8}, \frac{5}{12}$ | |

7. | $\frac{77}{225}$ | |

8. | $\frac{24}{115}$ | |

9. | $\frac{2}{495}, \frac{4}{165}$ | |

10. | $\frac{25}{39}$ | |

11. | $\frac{1}{2}, 500$ |

## ** Group (2011)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | Two fair dice are thrown. Draw a table to determine the set of all possible outcomes. Calculate the probability that the sum of the scores is less than $7 .$ Find also the probability that the product of the scores is even. $\mbox{ (5 marks)}$ | |

2. | A coin is tossed 4 times. Draw a tree diagram and list the possible outcomes. Find the probability that the number of heads is more than the number of tails. $\mbox{ (5 marks)}$ | |

3. | How many 2 -digit numerals can you form from $0,1,2,3$, without repeating any digit? Find the probability of a numeral which is divisible by $2 .$ $\mbox{ (5 marks)}$ | |

4. | Box A contains 4 pieces of paper numbered $1,2,3,4 .$ Box B contains 2 pieces of paper numbered 1,2 . One piece of paper is chosen at random from each box. Draw a tree diagram to list all possible outcomes of the experiment. Find the probability that the product of the two numbers chosen is at least 4 . Find also the probability that the sum of the two chosen numbers is equal to their product. $\mbox{ (5 marks)}$ | |

5. | A bag contains 15 discs of which 3 are white, 5 are red and 7 are blue. Three discs are to be drawn in succession with replacement. Calculate the probability that all the discs are of the same colour. $\mbox{ (5 marks)}$ | |

6. | If the probabilities that students $P$ and $Q$ will pass an examination are $\frac{5}{6}$ and $\frac{3}{4}$ respectively, find the probability that both $P$ and $Q$ will pass the examination. Find also the probability that at most one of $P$ and $Q$ will fail the examination. $\mbox{ (5 marks)}$ | |

7. | A die is rolled 360 times. Find the expected frequency of a factor of 6 and the expected frequency of a prime number. If all the scores obtained in these 360 trials are added together, what is the expected total score? $\mbox{ (5 marks)}$ |

#### ** Answer (2011)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | $\frac{5}{12} ; \frac{3}{4}$ | |

2. | $\frac{5}{16} $ | |

3. | $9 ; \frac{5}{9}$ | |

4. | $\frac{1}{2} ;\frac{1}{8}$ | |

5. | $\frac{11}{75}$ | |

6. | $\frac{5}{8}, \frac{23}{24}$ | |

7. | $240 ;180 ; 1260$ |

## ** Group (2010)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | A coin is tossed and a die is thrown.Head or tail and a number turns up are recorded each time.Draw a tree diagram and list possible outcomes.Find the probability that head and odd number turn up.$\text{ (5 marks)}$ | |

2. | A coin is tossed and a die is thrown.Head or tail and a number turn up are recorded.Draw a tree diagram and list possible outcomes.Find the probability that a head and an even number turn up.$\text{ (5 marks)}$ | |

3. | Draw a tree diagram to list all possible two-digit numerals which can be formed by using the digits $2,3,5$ and 6 without repetition.If one of these numerals is choser.at random, find the probability that it is divisible by $13.$ Find also the probability that it is either a prime number or a perfect square.$\text{ (5 marks)}$ | |

4. | A box contains 5 discs of 1 green, 2 yellow and 2 blue.A disc is chosen, the colour is recorded, and the disc is not replaced.Then another disc is chosen and the colour is recorded.Draw a tree diagram.Find the probability of choosing the same colours.Calculate the probability of choosing the different colours. | |

5. | Construct the table of outcomes for rolling two dice.Use it to find the probability that the total score is a prime number.Use it to calculate the probability that the total) score is greater than 10.$\text{ (5 marks)}$ | |

6. | A blue die and a black die are rolled.Construct a table of possible outcomes.Find the probability that the total score is a multiple of 3.Find also the probability that the product of the scores on the two dice is divisible by 4.$\text{ (5 marks)}$ | |

7. | If two dice are tossed, find the probability of getting a total of 10 or more, and calculate the probability of both dice showing the same number.$\text{ (5 marks)}$ | |

8. | The probabilities of students $A$ and $B$ to pass an examination are $\frac{2}{3}$ and $\frac{3}{4}$ respectively.Find the probability that at least one of $A$ and $B$ pass the examination.$\text{ (5 marks)}$ | |

9. | The probabilities of students $A$ and $B$ to fail an examination are $\frac{1}{3}$ and $\frac{1}{4}$ respectively.Find the probability that at least one of $A$ and $B$ pass the examination.$\text{ (5 marks)}$ | |

10. | Three groups of children consist of 3 boys and 1 girl, 2 boys and 2 girls, and 1 boy and 3 girls, respectively.If a child is chosen from each group, find the probability that 1 boy and 2 girls are chosen.$\text{ (5 marks)}$ | |

11. | $X$ and $Y$ are two independent events.The probability that the event $X$ will occur is twice the probability that the event $Y$ will occur and the probability that $Y$ will not occur is three times the probability that $X$ will not occur.Then find the probability that both $X$ and $Y$ will occur.$\text{ (5 marks)}$ |

#### ** Answer (2010)**

$\quad\;\,$ | ||
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1. | $\frac{1}{4}$ | |

2. | $\frac{1}{4}$ | |

3. | $\frac 14;\frac 13$ | |

4. | $\frac{1}{5} ; \frac{4}{5} \quad$ | |

5. | $\frac{5}{12} ; \frac{1}{12}$ | |

6. | $\frac{1}{3} ; \frac{5}{12}$ | |

7. | $\frac 16;\frac 16$ | |

8. | $\frac{11}{12}$ | |

9. | $\frac{11}{12}$ | |

10. | $\frac{13}{32}$ | |

11. | $\frac{8}{25}$ |

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