Binomial Probability

Binomial Probability If an event has only two possibilities and this event is repeated, then the

probability distribution is as follows;

Binomial Probability If an event has only two possibilities and this event is repeated, then the

probability distribution is as follows;

A

B

Binomial Probability If an event has only two possibilities and this event is repeated, then the

probability distribution is as follows;

A

B

p

q

Binomial Probability If 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 Probability If 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 Probability If 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 Probability If 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

AAB

ABA

ABB

BAA

BAB BBA

BBB

Binomial Probability If 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

AAB

ABA

ABB

BAA

BAB BBA

BBB

3p

qp2

qp2

2pq

qp2

2pq

2pq

3p

Binomial Probability If 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

AAB

ABA

ABB

BAA

BAB BBA

BBB

3p

qp2

qp2

2pq

qp2

2pq

2pq

3p

AAAA

AAAB AABA

AABB ABAA

ABAB ABBA

ABBB BAAA

BAAB BABA

BABB BBAA

BBAB BBBA

BBBB

Binomial Probability If 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

AAB

ABA

ABB

BAA

BAB BBA

BBB

3p

qp2

qp2

2pq

qp2

2pq

2pq

3p

AAAA

AAAB AABA

AABB ABAA

ABAB ABBA

ABBB BAAA

BAAB BABA

BABB BBAA

BBAB BBBA

BBBB

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

and times repeated isevent an If

k

XqXP

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

and times repeated isevent an If

k

XqXP

pXPn

kkn

k

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

7

7

5

3

5

27

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

7

7

5

3

5

27

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

b) 4 black balls?

43

4

7

5

3

5

24

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

b) 4 black balls?

43

4

7

5

3

5

24

CXP

7

43

4

7

5

32C

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

b) 4 black balls?

43

4

7

5

3

5

24

CXP

7

43

4

7

5

32C

15625

4536

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

b) 4 black balls?

43

4

7

5

3

5

24

CXP

7

43

4

7

5

32C

15625

4536

(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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

b) 4 black balls?

43

4

7

5

3

5

24

CXP

7

43

4

7

5

32C

15625

4536

(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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

b) 4 black balls?

43

4

7

5

3

5

24

CXP

7

43

4

7

5

32C

15625

4536

(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

3

5

10

3

10

73

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

b) 4 black balls?

43

4

7

5

3

5

24

CXP

7

43

4

7

5

32C

15625

4536

(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

3

5

10

3

10

73

CAP

5

32

3

5

10

37C

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

7

7

5

3

5

27

CXP

7

7

7

7

5

3C

78125

2187

b) 4 black balls?

43

4

7

5

3

5

24

CXP

7

43

4

7

5

32C

15625

4536

(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

3

5

10

3

10

73

CAP

5

32

3

5

10

37C

10000

1323

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

5

5

41

4

5

32

3

5

10

3

10

7

10

3

10

7

10

3

10

73

CCCXP

b) majority favour A?

50

5

5

41

4

5

32

3

5

10

3

10

7

10

3

10

7

10

3

10

73

CCCXP

5

5

5

54

4

532

3

5

10

33737 CCC

b) majority favour A?

50

5

5

41

4

5

32

3

5

10

3

10

7

10

3

10

7

10

3

10

73

CCCXP

5

5

5

54

4

532

3

5

10

33737 CCC

25000

4077

b) majority favour A?

50

5

5

41

4

5

32

3

5

10

3

10

7

10

3

10

7

10

3

10

73

CCCXP

5

5

5

54

4

532

3

5

10

33737 CCC

25000

4077

c) at most 2 favoured A?

b) majority favour A?

50

5

5

41

4

5

32

3

5

10

3

10

7

10

3

10

7

10

3

10

73

CCCXP

5

5

5

54

4

532

3

5

10

33737 CCC

25000

4077

c) at most 2 favoured A?

312 XPXP

b) majority favour A?

50

5

5

41

4

5

32

3

5

10

3

10

7

10

3

10

7

10

3

10

73

CCCXP

5

5

5

54

4

532

3

5

10

33737 CCC

25000

4077

c) at most 2 favoured A?

312 XPXP

25000

40771

b) majority favour A?

50

5

5

41

4

5

32

3

5

10

3

10

7

10

3

10

7

10

3

10

73

CCCXP

5

5

5

54

4

532

3

5

10

33737 CCC

25000

4077

c) at most 2 favoured A?

312 XPXP

25000

40771

25000

20923

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 3

2

(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 3

2

(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 3

2

(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

5

5

41

4

5

32

3

5

3

2

3

1

3

2

3

1

3

2

3

13

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 3

2

(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

5

5

41

4

5

32

3

5

3

2

3

1

3

2

3

1

3

2

3

13

CCCXP

5

5

5

54

4

53

3

5

3

222 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 3

2

(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

5

5

41

4

5

32

3

5

3

2

3

1

3

2

3

1

3

2

3

13

CCCXP

5

5

5

54

4

53

3

5

3

222 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

180

18

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

180

18

18 790102099018 CYP

dp) 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

180

18

18 790102099018 CYP

dp) 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

180

18

18 790102099018 CYP

dp) 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

180

18

18 790102099018 CYP

dp) 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

18

18171

17

18 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

180

18

18 790102099018 CYP

dp) 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

180

18

18171

17

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

0.01

P

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

0.01

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

2 green 0.1 0.1

0.01

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

18 220

22 0.9 0.1P X C

2 green 0.1 0.1

0.01

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

18 220

22 0.9 0.1P X C

0.2851

0.285 (to 3 dp)

2 green 0.1 0.1

0.01

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 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

0.3230

0 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

0.3230

0 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