Binomial Probability

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

p

q

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

p

q

AA

AB

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

p

q

AA

AB

BA

BB

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

p

q

AA

AB

BA

BB

2p

pq

pq

2q

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

p

q

AA

AB

BA

BB

2p

pq

pq

2q

AAA

AABABA

ABB

BAA

BABBBA

BBB

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

p

q

AA

AB

BA

BB

2p

pq

pq

2q

AAA

AABABA

ABB

BAA

BABBBA

BBB

3p

qp2

qp2

2pq

qp2

2pq 2pq

3p

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

p

q

AA

AB

BA

BB

2p

pq

pq

2q

AAA

AABABA

ABB

BAA

BABBBA

BBB

3p

qp2

qp2

2pq

qp2

2pq 2pq

3p

AAAAAAAB

AABAAABBABAAABABABBA

ABBBBAAA

BAABBABA

BABBBBAABBABBBBABBBB

Binomial ProbabilityIf an event has only two possibilities and this event is repeated, then the probability distribution is as follows;

A

B

p

q

AA

AB

BA

BB

2p

pq

pq

2q

AAA

AABABA

ABB

BAA

BABBBA

BBB

3p

qp2

qp2

2pq

qp2

2pq 2pq

3p

AAAAAAAB

AABAAABBABAAABABABBA

ABBBBAAA

BAABBABA

BABBBBAABBABBBBABBBB

4p qp3

qp3

22qp qp3

22qp 22qp 3pq qp3

22qp 22qp 3pq 22qp 3pq 3pq 4q

1 Event

1 Event pAP

qBP

1 Event pAP

qBP

2 Events

1 Event pAP

qBP

2 Events 2pAAP pqAandBP 2 2qBBP

1 Event pAP

qBP

2 Events 2pAAP pqAandBP 2 2qBBP

3 Events

1 Event pAP

qBP

2 Events 2pAAP pqAandBP 2 2qBBP

3 Events 3pAAAP qpAandBP 232 232 pqBAandP 3qBBBP

1 Event pAP

qBP

2 Events 2pAAP pqAandBP 2 2qBBP

3 Events 3pAAAP qpAandBP 232 232 pqBAandP 3qBBBP

4 Events

1 Event pAP

qBP

2 Events 2pAAP pqAandBP 2 2qBBP

3 Events 3pAAAP qpAandBP 232 232 pqBAandP 3qBBBP

4 Events 4pAAAAP qpAandBP 343 22622 qpBAandP 343 pqBAandP 4qBBBBP

1 Event pAP

qBP

2 Events 2pAAP pqAandBP 2 2qBBP

3 Events 3pAAAP qpAandBP 232 232 pqBAandP 3qBBBP

4 Events 4pAAAAP qpAandBP 343 22622 qpBAandP 343 pqBAandP 4qBBBBP

is; timesexactly occur willy that probabilit then the and

andtimesrepeatedisevent an If

kXqXP

pXPn

1 Event pAP

qBP

2 Events 2pAAP pqAandBP 2 2qBBP

3 Events 3pAAAP qpAandBP 232 232 pqBAandP 3qBBBP

4 Events 4pAAAAP qpAandBP 343 22622 qpBAandP 343 pqBAandP 4qBBBBP

is; timesexactly occur willy that probabilit then the and

andtimesrepeatedisevent an If

kXqXP

pXPn

kknk

n pqCkXP

Note: X = k, means X will occur exactly k times

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

Let X be the number of black balls drawn

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

Let X be the number of black balls drawn

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

Let X be the number of black balls drawn

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?Let X be the number of black balls drawn

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?

43

47

53

524

CXP

Let X be the number of black balls drawn

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?

43

47

53

524

CXP

7

434

7

532C

Let X be the number of black balls drawn

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?

43

47

53

524

CXP

7

434

7

532C

156254536

Let X be the number of black balls drawn

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?

43

47

53

524

CXP

7

434

7

532C

156254536

(ii) At an election 30% of voters favoured Party A.If at random an interviewer selects 5 voters, what is the probability that;

a) 3 favoured Party A?

Let X be the number of black balls drawn

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?

43

47

53

524

CXP

7

434

7

532C

156254536

(ii) At an election 30% of voters favoured Party A.If at random an interviewer selects 5 voters, what is the probability that;

a) 3 favoured Party A?

Let X be the number of black balls drawn

Let X be the number favouring Party A

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?

43

47

53

524

CXP

7

434

7

532C

156254536

(ii) At an election 30% of voters favoured Party A.If at random an interviewer selects 5 voters, what is the probability that;

a) 3 favoured Party A? 32

35

103

1073

CAP

Let X be the number of black balls drawn

Let X be the number favouring Party A

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?

43

47

53

524

CXP

7

434

7

532C

156254536

(ii) At an election 30% of voters favoured Party A.If at random an interviewer selects 5 voters, what is the probability that;

a) 3 favoured Party A? 32

35

103

1073

CAP

5

323

5

1037C

Let X be the number of black balls drawn

Let X be the number favouring Party A

e.g.(i) A bag contains 30 black balls and 20 white balls.Seven drawings are made (with replacement), what is the probability of drawing;

a) All black balls?

70

77

53

527

CXP

7

77

7

53C

781252187

b) 4 black balls?

43

47

53

524

CXP

7

434

7

532C

156254536

(ii) At an election 30% of voters favoured Party A.If at random an interviewer selects 5 voters, what is the probability that;

a) 3 favoured Party A? 32

35

103

1073

CAP

5

323

5

1037C

100001323

Let X be the number of black balls drawn

Let X be the number favouring Party A

b) majority favour A?

b) majority favour A?

50

55

41

45

32

35

103

107

103

107

103

1073

CCCXP

b) majority favour A?

50

55

41

45

32

35

103

107

103

107

103

1073

CCCXP

5

55

544

5323

5

1033737 CCC

b) majority favour A?

50

55

41

45

32

35

103

107

103

107

103

1073

CCCXP

5

55

544

5323

5

1033737 CCC

250004077

b) majority favour A?

50

55

41

45

32

35

103

107

103

107

103

1073

CCCXP

5

55

544

5323

5

1033737 CCC

250004077

c) at most 2 favoured A?

b) majority favour A?

50

55

41

45

32

35

103

107

103

107

103

1073

CCCXP

5

55

544

5323

5

1033737 CCC

250004077

c) at most 2 favoured A?

312 XPXP

b) majority favour A?

50

55

41

45

32

35

103

107

103

107

103

1073

CCCXP

5

55

544

5323

5

1033737 CCC

250004077

c) at most 2 favoured A?

312 XPXP

2500040771

b) majority favour A?

50

55

41

45

32

35

103

107

103

107

103

1073

CCCXP

5

55

544

5323

5

1033737 CCC

250004077

c) at most 2 favoured A?

312 XPXP

2500040771

2500020923

2005 Extension 1 HSC Q6a)There are five matches on each weekend of a football season. Megan takes part in a competition in which she earns 1 point if she picks more than half of the winning teams for a weekend, and zero points otherwise.

The probability that Megan correctly picks the team that wins in any given match is

32

(i) Show that the probability that Megan earns one point for a given weekend is , correct to four decimal places. 79010

2005 Extension 1 HSC Q6a)There are five matches on each weekend of a football season. Megan takes part in a competition in which she earns 1 point if she picks more than half of the winning teams for a weekend, and zero points otherwise.

The probability that Megan correctly picks the team that wins in any given match is

32

(i) Show that the probability that Megan earns one point for a given weekend is , correct to four decimal places. 79010

Let X be the number of matches picked correctly

2005 Extension 1 HSC Q6a)There are five matches on each weekend of a football season. Megan takes part in a competition in which she earns 1 point if she picks more than half of the winning teams for a weekend, and zero points otherwise.

The probability that Megan correctly picks the team that wins in any given match is

32

(i) Show that the probability that Megan earns one point for a given weekend is , correct to four decimal places. 79010

Let X be the number of matches picked correctly

50

55

41

45

32

35

32

31

32

31

32

313

CCCXP

2005 Extension 1 HSC Q6a)There are five matches on each weekend of a football season. Megan takes part in a competition in which she earns 1 point if she picks more than half of the winning teams for a weekend, and zero points otherwise.

The probability that Megan correctly picks the team that wins in any given match is

32

(i) Show that the probability that Megan earns one point for a given weekend is , correct to four decimal places. 79010

Let X be the number of matches picked correctly

50

55

41

45

32

35

32

31

32

31

32

313

CCCXP

5

55

544

533

5

3222 CCC

2005 Extension 1 HSC Q6a)There are five matches on each weekend of a football season. Megan takes part in a competition in which she earns 1 point if she picks more than half of the winning teams for a weekend, and zero points otherwise.

The probability that Megan correctly picks the team that wins in any given match is

32

(i) Show that the probability that Megan earns one point for a given weekend is , correct to four decimal places. 79010

Let X be the number of matches picked correctly

50

55

41

45

32

35

32

31

32

31

32

313

CCCXP

5

55

544

533

5

3222 CCC

79010

(ii) Hence find the probability that Megan earns one point every week of the eighteen week season. Give your answer correct to two decimal places.

(ii) Hence find the probability that Megan earns one point every week of the eighteen week season. Give your answer correct to two decimal places.

Let Y be the number of weeks Megan earns a point

(ii) Hence find the probability that Megan earns one point every week of the eighteen week season. Give your answer correct to two decimal places.

Let Y be the number of weeks Megan earns a point

18018

18 790102099018 CYP

(ii) Hence find the probability that Megan earns one point every week of the eighteen week season. Give your answer correct to two decimal places.

Let Y be the number of weeks Megan earns a point

18018

18 790102099018 CYPdp)2 (to 010

(ii) Hence find the probability that Megan earns one point every week of the eighteen week season. Give your answer correct to two decimal places.

Let Y be the number of weeks Megan earns a point

18018

18 790102099018 CYPdp)2 (to 010

(iii) Find the probability that Megan earns at most 16 points during the eighteen week season. Give your answer correct to two decimal places.

(ii) Hence find the probability that Megan earns one point every week of the eighteen week season. Give your answer correct to two decimal places.

Let Y be the number of weeks Megan earns a point

18018

18 790102099018 CYPdp)2 (to 010

(iii) Find the probability that Megan earns at most 16 points during the eighteen week season. Give your answer correct to two decimal places.

17116 YPYP

(ii) Hence find the probability that Megan earns one point every week of the eighteen week season. Give your answer correct to two decimal places.

Let Y be the number of weeks Megan earns a point

18018

18 790102099018 CYPdp)2 (to 010

(iii) Find the probability that Megan earns at most 16 points during the eighteen week season. Give your answer correct to two decimal places.

17116 YPYP 180

1818171

1718 790102099079010209901 CC

(ii) Hence find the probability that Megan earns one point every week of the eighteen week season. Give your answer correct to two decimal places.

Let Y be the number of weeks Megan earns a point

18018

18 790102099018 CYPdp)2 (to 010

(iii) Find the probability that Megan earns at most 16 points during the eighteen week season. Give your answer correct to two decimal places.

17116 YPYP

dp)2 (to 920

18018

1817117

18 790102099079010209901 CC

2007 Extension 1 HSC Q4a)In a large city, 10% of the population has green eyes.(i) What is the probability that two randomly chosen people have

green eyes?

2007 Extension 1 HSC Q4a)In a large city, 10% of the population has green eyes.(i) What is the probability that two randomly chosen people have

green eyes? 2 green 0.1 0.1

0.01P

2007 Extension 1 HSC Q4a)In a large city, 10% of the population has green eyes.(i) What is the probability that two randomly chosen people have

green eyes? 2 green 0.1 0.1

0.01P

(ii) What is the probability that exactly two of a group of 20 randomly

chosen people have green eyes? Give your answer correct to three decimal eyes.

2007 Extension 1 HSC Q4a)In a large city, 10% of the population has green eyes.(i) What is the probability that two randomly chosen people have

green eyes?

Let X be the number of people with green eyes

2 green 0.1 0.10.01

P

(ii) What is the probability that exactly two of a group of 20 randomly

chosen people have green eyes? Give your answer correct to three decimal eyes.

2007 Extension 1 HSC Q4a)In a large city, 10% of the population has green eyes.(i) What is the probability that two randomly chosen people have

green eyes?

Let X be the number of people with green eyes

18 22022 0.9 0.1P X C

2 green 0.1 0.10.01

P

(ii) What is the probability that exactly two of a group of 20 randomly

chosen people have green eyes? Give your answer correct to three decimal eyes.

2007 Extension 1 HSC Q4a)In a large city, 10% of the population has green eyes.(i) What is the probability that two randomly chosen people have

green eyes?

Let X be the number of people with green eyes

18 22022 0.9 0.1P X C

0.28510.285 (to 3 dp)

2 green 0.1 0.10.01

P

(ii) What is the probability that exactly two of a group of 20 randomly

chosen people have green eyes? Give your answer correct to three decimal eyes.

(iii) What is the probability that more than two of a group of 20 randomly chosen people have green eyes? Give your answer correct to two decimal places.

(iii) What is the probability that more than two of a group of 20 randomly chosen people have green eyes? Give your answer correct to two decimal places.

2 1 2P X P X

(iii) What is the probability that more than two of a group of 20 randomly chosen people have green eyes? Give your answer correct to two decimal places.

2 1 2P X P X

18 2 19 1 20 020 20 202 1 01 0 9 0 1 0 9 0 1 0 9 0 1C C C

(iii) What is the probability that more than two of a group of 20 randomly chosen people have green eyes? Give your answer correct to two decimal places.

2 1 2P X P X

0.32300 32 (to 2 dp)

18 2 19 1 20 020 20 20

2 1 01 0 9 0 1 0 9 0 1 0 9 0 1C C C

(iii) What is the probability that more than two of a group of 20 randomly chosen people have green eyes? Give your answer correct to two decimal places.

2 1 2P X P X

0.32300 32 (to 2 dp)

18 2 19 1 20 020 20 20

2 1 01 0 9 0 1 0 9 0 1 0 9 0 1C C C

Exercise 10J; 1 to 24 even