11x1 t05 04 probability & counting techniques (2011)

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Probability & Counting Techniques Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row. What is the probability that the four children are allocated seats next to each other? 2007 Extension 1 HSC Q5b)

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Page 1: 11X1 T05 04 probability & counting techniques (2011)

Probability & Counting Techniques

Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row.

What is the probability that the four children are allocated seats next to each other?

2007 Extension 1 HSC Q5b)

Page 2: 11X1 T05 04 probability & counting techniques (2011)

Probability & Counting Techniques

Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row.

What is the probability that the four children are allocated seats next to each other?

3!4!(children sit next to each other)6!

P

2007 Extension 1 HSC Q5b)

Page 3: 11X1 T05 04 probability & counting techniques (2011)

Probability & Counting Techniques

Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row.

What is the probability that the four children are allocated seats next to each other?

3!4!(children sit next to each other)6!

P

2007 Extension 1 HSC Q5b)

ways of arranging 6 people

Page 4: 11X1 T05 04 probability & counting techniques (2011)

Probability & Counting Techniques

Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row.

What is the probability that the four children are allocated seats next to each other?

3!4!(children sit next to each other)6!

P

2007 Extension 1 HSC Q5b)

ways of arranging 6 people

ways of arranging 3 objectsi.e 2 adults + 1 group of 4 children

Page 5: 11X1 T05 04 probability & counting techniques (2011)

Probability & Counting Techniques

Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row.

What is the probability that the four children are allocated seats next to each other?

3!4!(children sit next to each other)6!

P

2007 Extension 1 HSC Q5b)

ways of arranging 6 people

ways of arranging 3 objectsi.e 2 adults + 1 group of 4 children ways of arranging 4 children

Page 6: 11X1 T05 04 probability & counting techniques (2011)

Probability & Counting Techniques

Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row.

What is the probability that the four children are allocated seats next to each other?

3!4!(children sit next to each other)6!

P

2007 Extension 1 HSC Q5b)

ways of arranging 6 people

ways of arranging 3 objectsi.e 2 adults + 1 group of 4 children ways of arranging 4 children

15

Page 7: 11X1 T05 04 probability & counting techniques (2011)

2007 Extension 2 HSC Q5a)A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected at random without replacement.

(i) Calculate the probability that exactly three of the selected marbles are red. Give your answer correct to two decimal places.

Page 8: 11X1 T05 04 probability & counting techniques (2011)

2007 Extension 2 HSC Q5a)A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected at random without replacement.

(i) Calculate the probability that exactly three of the selected marbles are red. Give your answer correct to two decimal places.

12 123 324

6

(3 red) C CPC

Page 9: 11X1 T05 04 probability & counting techniques (2011)

2007 Extension 2 HSC Q5a)A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected at random without replacement.

(i) Calculate the probability that exactly three of the selected marbles are red. Give your answer correct to two decimal places.

12 123 324

6

(3 red) C CPC

0.35950.36 (to 2 dp)

Page 10: 11X1 T05 04 probability & counting techniques (2011)

2007 Extension 2 HSC Q5a)A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected at random without replacement.

(i) Calculate the probability that exactly three of the selected marbles are red. Give your answer correct to two decimal places.

12 123 324

6

(3 red) C CPC

0.35950.36 (to 2 dp)

(ii) Hence, or otherwise, calculate the probability that more than three of the selected marbles are red. Give your answer correct to two decimal places.

Page 11: 11X1 T05 04 probability & counting techniques (2011)

2007 Extension 2 HSC Q5a)A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected at random without replacement.

(i) Calculate the probability that exactly three of the selected marbles are red. Give your answer correct to two decimal places.

12 123 324

6

(3 red) C CPC

0.35950.36 (to 2 dp)

(ii) Hence, or otherwise, calculate the probability that more than three of the selected marbles are red. Give your answer correct to two decimal places.

( 3 red) (4 red) (5 red)+ (6 red)P P P P

Page 12: 11X1 T05 04 probability & counting techniques (2011)

2007 Extension 2 HSC Q5a)A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected at random without replacement.

(i) Calculate the probability that exactly three of the selected marbles are red. Give your answer correct to two decimal places.

12 123 324

6

(3 red) C CPC

0.35950.36 (to 2 dp)

(ii) Hence, or otherwise, calculate the probability that more than three of the selected marbles are red. Give your answer correct to two decimal places.

( 3 red) (4 red) (5 red)+ (6 red)P P P P 12 12 12 12 12 12

4 2 5 1 6 024

6

C C C C C CC

Page 13: 11X1 T05 04 probability & counting techniques (2011)

2007 Extension 2 HSC Q5a)A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected at random without replacement.

(i) Calculate the probability that exactly three of the selected marbles are red. Give your answer correct to two decimal places.

12 123 324

6

(3 red) C CPC

0.35950.36 (to 2 dp)

(ii) Hence, or otherwise, calculate the probability that more than three of the selected marbles are red. Give your answer correct to two decimal places.

( 3 red) (4 red) (5 red)+ (6 red)P P P P 12 12 12 12 12 12

4 2 5 1 6 024

6

C C C C C CC

0.32020.32 (to 2 dp)

Page 14: 11X1 T05 04 probability & counting techniques (2011)

OR( 3 red) 1 (3 red) ( 3 red)P P P

Page 15: 11X1 T05 04 probability & counting techniques (2011)

OR( 3 red) 1 (3 red) ( 3 red)P P P

1 (3 red) ( 3 yellow)P P

Page 16: 11X1 T05 04 probability & counting techniques (2011)

OR( 3 red) 1 (3 red) ( 3 red)P P P

1 (3 red) ( 3 yellow)P P

1 (3 red) ( 3 red)P P

Page 17: 11X1 T05 04 probability & counting techniques (2011)

OR( 3 red) 1 (3 red) ( 3 red)P P P

1 (3 red) ( 3 yellow)P P

1 (3 red) ( 3 red)P P

2 ( 3 red) 1 (3 red)P P

Page 18: 11X1 T05 04 probability & counting techniques (2011)

OR( 3 red) 1 (3 red) ( 3 red)P P P

1 (3 red) ( 3 yellow)P P

1 (3 red) ( 3 red)P P

2 ( 3 red) 1 (3 red)P P

1( 3 red) 1 (3 red)2

P P

Page 19: 11X1 T05 04 probability & counting techniques (2011)

OR( 3 red) 1 (3 red) ( 3 red)P P P

1 (3 red) ( 3 yellow)P P

1 (3 red) ( 3 red)P P

2 ( 3 red) 1 (3 red)P P

1( 3 red) 1 (3 red)2

P P

1 1 0.35952

Page 20: 11X1 T05 04 probability & counting techniques (2011)

OR( 3 red) 1 (3 red) ( 3 red)P P P

1 (3 red) ( 3 yellow)P P

1 (3 red) ( 3 red)P P

2 ( 3 red) 1 (3 red)P P

1( 3 red) 1 (3 red)2

P P

1 1 0.35952

0.32020.32 (to 2 dp)

Page 21: 11X1 T05 04 probability & counting techniques (2011)

2006 Extension 2 HSC Q5d)In a chess match between the Home team and the Away team, a game is played on board 1, board 2, board 3 and board 4.

On each board, the probability that the Home team wins is 0.2, the probability of a draw is 0.6 and the probability that the Home team loses is 0.2.

The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 2, draws on board 2, loses on board 3 and draws on board 4, the result is recorded as WDLD.

Page 22: 11X1 T05 04 probability & counting techniques (2011)

2006 Extension 2 HSC Q5d)In a chess match between the Home team and the Away team, a game is played on board 1, board 2, board 3 and board 4.

On each board, the probability that the Home team wins is 0.2, the probability of a draw is 0.6 and the probability that the Home team loses is 0.2.

The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 2, draws on board 2, loses on board 3 and draws on board 4, the result is recorded as WDLD.

(i) How many different recordings are possible?

Page 23: 11X1 T05 04 probability & counting techniques (2011)

2006 Extension 2 HSC Q5d)In a chess match between the Home team and the Away team, a game is played on board 1, board 2, board 3 and board 4.

On each board, the probability that the Home team wins is 0.2, the probability of a draw is 0.6 and the probability that the Home team loses is 0.2.

The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 2, draws on board 2, loses on board 3 and draws on board 4, the result is recorded as WDLD.

(i) How many different recordings are possible?Recordings 3 3 3 3

Page 24: 11X1 T05 04 probability & counting techniques (2011)

2006 Extension 2 HSC Q5d)In a chess match between the Home team and the Away team, a game is played on board 1, board 2, board 3 and board 4.

On each board, the probability that the Home team wins is 0.2, the probability of a draw is 0.6 and the probability that the Home team loses is 0.2.

The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 2, draws on board 2, loses on board 3 and draws on board 4, the result is recorded as WDLD.

(i) How many different recordings are possible?Recordings 3 3 3 3

81

Page 25: 11X1 T05 04 probability & counting techniques (2011)

2006 Extension 2 HSC Q5d)In a chess match between the Home team and the Away team, a game is played on board 1, board 2, board 3 and board 4.

On each board, the probability that the Home team wins is 0.2, the probability of a draw is 0.6 and the probability that the Home team loses is 0.2.

The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 2, draws on board 2, loses on board 3 and draws on board 4, the result is recorded as WDLD.

(i) How many different recordings are possible?Recordings 3 3 3 3

81(ii) Calculate the probability of the result which is recorded as WDLD.

Page 26: 11X1 T05 04 probability & counting techniques (2011)

2006 Extension 2 HSC Q5d)In a chess match between the Home team and the Away team, a game is played on board 1, board 2, board 3 and board 4.

On each board, the probability that the Home team wins is 0.2, the probability of a draw is 0.6 and the probability that the Home team loses is 0.2.

The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 2, draws on board 2, loses on board 3 and draws on board 4, the result is recorded as WDLD.

(i) How many different recordings are possible?Recordings 3 3 3 3

81(ii) Calculate the probability of the result which is recorded as WDLD.

WDLD 0.2 0.6 0.2 0.6P

Page 27: 11X1 T05 04 probability & counting techniques (2011)

2006 Extension 2 HSC Q5d)In a chess match between the Home team and the Away team, a game is played on board 1, board 2, board 3 and board 4.

On each board, the probability that the Home team wins is 0.2, the probability of a draw is 0.6 and the probability that the Home team loses is 0.2.

The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 2, draws on board 2, loses on board 3 and draws on board 4, the result is recorded as WDLD.

(i) How many different recordings are possible?Recordings 3 3 3 3

81(ii) Calculate the probability of the result which is recorded as WDLD.

WDLD 0.2 0.6 0.2 0.6P 0.144

Page 28: 11X1 T05 04 probability & counting techniques (2011)

(iii) Teams score 1 point for each game won, a point for each game drawn and 0 points for each game lost.What is the probability that the Home team scores more points than the Away team?

12

Page 29: 11X1 T05 04 probability & counting techniques (2011)

(iii) Teams score 1 point for each game won, a point for each game drawn and 0 points for each game lost.What is the probability that the Home team scores more points than the Away team?

12

first calculate probability of equal points

Page 30: 11X1 T05 04 probability & counting techniques (2011)

(iii) Teams score 1 point for each game won, a point for each game drawn and 0 points for each game lost.What is the probability that the Home team scores more points than the Away team?

12

first calculate probability of equal points

44 draws 0.60.1296

P

Page 31: 11X1 T05 04 probability & counting techniques (2011)

(iii) Teams score 1 point for each game won, a point for each game drawn and 0 points for each game lost.What is the probability that the Home team scores more points than the Away team?

12

first calculate probability of equal points

44 draws 0.60.1296

P

2 2 4!2 wins, 2 losses 0.2 0.22!2!

0.0096

P

Page 32: 11X1 T05 04 probability & counting techniques (2011)

(iii) Teams score 1 point for each game won, a point for each game drawn and 0 points for each game lost.What is the probability that the Home team scores more points than the Away team?

12

first calculate probability of equal points

44 draws 0.60.1296

P

2 2 4!2 wins, 2 losses 0.2 0.22!2!

0.0096

P

ways of arranging WWLL

Page 33: 11X1 T05 04 probability & counting techniques (2011)

(iii) Teams score 1 point for each game won, a point for each game drawn and 0 points for each game lost.What is the probability that the Home team scores more points than the Away team?

12

first calculate probability of equal points

44 draws 0.60.1296

P

2 2 4!2 wins, 2 losses 0.2 0.22!2!

0.0096

P

ways of arranging WWLL

2 4!1 win, 1 loss, 2 draws 0.2 0.2 0.62!

0.1728

P

Page 34: 11X1 T05 04 probability & counting techniques (2011)

(iii) Teams score 1 point for each game won, a point for each game drawn and 0 points for each game lost.What is the probability that the Home team scores more points than the Away team?

12

first calculate probability of equal points

44 draws 0.60.1296

P

2 2 4!2 wins, 2 losses 0.2 0.22!2!

0.0096

P

ways of arranging WWLL

2 4!1 win, 1 loss, 2 draws 0.2 0.2 0.62!

0.1728

P

ways of arranging WLDD

Page 35: 11X1 T05 04 probability & counting techniques (2011)

(iii) Teams score 1 point for each game won, a point for each game drawn and 0 points for each game lost.What is the probability that the Home team scores more points than the Away team?

12

first calculate probability of equal points

44 draws 0.60.1296

P

2 2 4!2 wins, 2 losses 0.2 0.22!2!

0.0096

P

ways of arranging WWLL

2 4!1 win, 1 loss, 2 draws 0.2 0.2 0.62!

0.1728

P

ways of arranging WLDD

equal points 0.1296 0.0096 0.17280.312

P

Page 36: 11X1 T05 04 probability & counting techniques (2011)

unequal points 1 0.3120.688

P

Page 37: 11X1 T05 04 probability & counting techniques (2011)

unequal points 1 0.3120.688

P

As the probabilities are equally likely for the Home and Away teams, then either the Home team has more points or the Away team has more points.

Page 38: 11X1 T05 04 probability & counting techniques (2011)

unequal points 1 0.3120.688

P

As the probabilities are equally likely for the Home and Away teams, then either the Home team has more points or the Away team has more points.

1Home team more points unequal points2

P P

Page 39: 11X1 T05 04 probability & counting techniques (2011)

unequal points 1 0.3120.688

P

As the probabilities are equally likely for the Home and Away teams, then either the Home team has more points or the Away team has more points.

1Home team more points unequal points2

P P

1 0.68820.344

Page 40: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

Page 41: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?

Page 42: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P

Page 43: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P it is the same as saying; “what is the probability of the first number being >4”

Page 44: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P

23

it is the same as saying; “what is the probability of the first number being >4”

Page 45: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P

23

(ii) What is the probability that the digits are drawn in descending order?

it is the same as saying; “what is the probability of the first number being >4”

Page 46: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P

23

(ii) What is the probability that the digits are drawn in descending order?total arrangements of 3 digits 3!

it is the same as saying; “what is the probability of the first number being >4”

Page 47: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P

23

(ii) What is the probability that the digits are drawn in descending order?total arrangements of 3 digits 3!

6

it is the same as saying; “what is the probability of the first number being >4”

Page 48: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P

23

(ii) What is the probability that the digits are drawn in descending order?total arrangements of 3 digits 3!

6

it is the same as saying; “what is the probability of the first number being >4”

Only one arrangement will be in descending order

Page 49: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P

23

(ii) What is the probability that the digits are drawn in descending order?total arrangements of 3 digits 3!

6

it is the same as saying; “what is the probability of the first number being >4”

Only one arrangement will be in descending order

1descending order6

P

Page 50: 11X1 T05 04 probability & counting techniques (2011)

2002 Extension 2 HSC Q4c)From a pack of nine cards numbered 1, 2, 3, …, 9, three cards are drawn at random and laid on a table from left to right.

(i) What is the probability that the number exceeds 400?6( 400)9

P

23

(ii) What is the probability that the digits are drawn in descending order?total arrangements of 3 digits 3!

6

it is the same as saying; “what is the probability of the first number being >4”

Only one arrangement will be in descending order

1descending order6

P Exercise 10H; odd