2. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows;
3. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows;
A
B
4. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows;
A p
Bq
5. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows;
AA
A p
AB
Bq
6. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows;
AA
A p
AB
BA
Bq
BB
7. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows;
AA p 2
A p
AB pq
BA pq
Bq
BB q 2
8. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows; AAA
AA p 2
AAB
A p ABA
AB pq
ABB
BAA
BA pq
BAB
Bq BBA
BB q 2
BBB
9. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows; AAA p 3
AA p 2
AAB p q
2
A p ABA p 2 q
AB pq
ABB pq 2
BAA p 2 q
BA pq
BAB pq
2
Bq BBA pq 2
BB q 2
BBB p
3
10. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows; AAA p 3 AAAA
AAAB
AA p
2
AABA
AAB p q
2
AABB
A p ABA p q
2
ABAA
AB pq ABAB
ABBA
ABB pq 2
ABBB
BAAA
BAA p 2 q BAAB
BA pq BABA
BAB pq
2
BABB
Bq BBA pq 2 BBAA
BB q 2 BBAB
BBB p
3 BBBA
BBBB
11. Binomial Probability
If an event has only two possibilities and this event is repeated, then the
probability distribution is as follows; AAA p 3 AAAA p 4
AAAB p 3q
AA p 2 AABA p 3q
AAB p q AABB p 2 q 2
2
A p ABA p 2 q ABAA p 3q
AB pq ABAB p 2 q 2
ABB pq 2 p q
ABBA 2 2
ABBB pq 3
BAA p q
BAAA 3
p q
BAAB p 2 q 2
2
BA pq BABA p 2 q 2
BAB pq BABB pq
2 3
Bq BBA pq 2 BBAA p q
2 2
BB q 2
BBAB pq 3
BBB p pqq
3 BBBA 3
4
BBBB
15. 1 Event 2 Events
P A p P AA p 2
P B q P AandB 2 pq
PBB q 2
16. 1 Event 2 Events 3 Events
P A p P AA p 2
P B q P AandB 2 pq
PBB q 2
17. 1 Event 2 Events 3 Events
P A p P AA p 2 P AAA p 3
P B q P AandB 2 pq P2 AandB 3 p 2 q
PBB q 2 P Aand 2 B 3 pq 2
PBBB q 3
18. 1 Event 2 Events 3 Events
P A p P AA p 2 P AAA p 3
P B q P AandB 2 pq P2 AandB 3 p 2 q
PBB q 2 P Aand 2 B 3 pq 2
PBBB q 3
4 Events
19. 1 Event 2 Events 3 Events
P A p P AA p 2 P AAA p 3
P B q P AandB 2 pq P2 AandB 3 p 2 q
PBB q 2 P Aand 2 B 3 pq 2
PBBB q 3
4 Events
P AAAA p 4
P3 AandB 4 p 3q
P2 Aand 2 B 6 p 2 q 2
P Aand 3B 4 pq 3
PBBBB q 4
20. 1 Event 2 Events 3 Events
P A p P AA p 2 P AAA p 3
P B q P AandB 2 pq P2 AandB 3 p 2 q
PBB q 2 P Aand 2 B 3 pq 2
PBBB q 3
4 Events If an event is repeated n times and P X p
P AAAA p 4 and P X q then the probabilit y that X will
P3 AandB 4 p 3q occur exactly k times is;
P2 Aand 2 B 6 p 2 q 2
P Aand 3B 4 pq 3
PBBBB q 4
21. 1 Event 2 Events 3 Events
P A p P AA p 2 P AAA p 3
P B q P AandB 2 pq P2 AandB 3 p 2 q
PBB q 2 P Aand 2 B 3 pq 2
PBBB q 3
4 Events If an event is repeated n times and P X p
P AAAA p 4 and P X q then the probabilit y that X will
P3 AandB 4 p 3q occur exactly k times is;
P2 Aand 2 B 6 p 2 q 2
P X k nCk q nk p k
P Aand 3B 4 pq 3
PBBBB q 4 Note: X = k, means X will occur exactly k times
22. 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?
23. 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?
0 7
P X 7 7C7
2 3
5 5
24. 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; Let X be the number of black balls drawn
a) All black balls?
0 7
P X 7 7C7
2 3
5 5
25. 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; Let X be the number of black balls drawn
a) All black balls?
0 7
P X 7 7C7
2 3
5 5
7
C7 37
7
5
26. 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; Let X be the number of black balls drawn
a) All black balls?
0 7
P X 7 7C7
2 3
5 5
7
C7 37
7
5
2187
78125
27. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7
P X 7 7C7
2 3
5 5
7
C7 37
7
5
2187
78125
28. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7 3 4
P X 7 7C7
2 3
P X 4 7C4
2 3
5 5 5 5
7
C7 37
7
5
2187
78125
29. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7 3 4
P X 7 7C7
2 3
P X 4 7C4
2 3
5 5 5 5
7
C7 37 7
C4 2334
7
5 57
2187
78125
30. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7 3 4
P X 7 7C7
2 3
P X 4 7C4
2 3
5 5 5 5
7
C7 37 7
C4 2334
7
5 57
2187 4536
78125 15625
31. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7 3 4
P X 7 7C7
2 3
P X 4 7C4
2 3
5 5 5 5
7
C7 37 7
C4 2334
7
5 57
2187 4536
78125 15625
(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. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7 3 4
P X 7 7C7
2 3
P X 4 7C4
2 3
5 5 5 5
7
C7 37 7
C4 2334
7
5 57
2187 4536
78125 15625
(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
favouring Party A
33. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7 3 4
P X 7 7C7
2 3
P X 4 7C4
2 3
5 5 5 5
7
C7 37 7
C4 2334
7
5 57
2187 4536
78125 15625
(ii) At an election 30% of voters favoured Party A.
If at random an interviewer selects 5 voters, what is the
probability that; 2 3
P3 A5C3
7 3
a) 3 favoured Party A?
10 10
Let X be the number
favouring Party A
34. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7 3 4
P X 7 7C7
2 3
P X 4 7C4
2 3
5 5 5 5
7
C7 37 7
C4 2334
7
5 57
2187 4536
78125 15625
(ii) At an election 30% of voters favoured Party A.
If at random an interviewer selects 5 voters, what is the
probability that; 2 3
P3 A5C3
7 3
a) 3 favoured Party A?
10 10
Let X be the number 5
C3 7 233
favouring Party A
105
35. 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; Let X be the number of black balls drawn
a) All black balls? b) 4 black balls?
0 7 3 4
P X 7 7C7
2 3
P X 4 7C4
2 3
5 5 5 5
7
C7 37 7
C4 2334
7
5 57
2187 4536
78125 15625
(ii) At an election 30% of voters favoured Party A.
If at random an interviewer selects 5 voters, what is the
probability that; 2 3
P3 A5C3
7 3
a) 3 favoured Party A?
10 10
Let X be the number 5
C3 7 233 1323
favouring Party A 5
10 10000
44. 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 2
3
(i) Show that the probability that Megan earns one point for a
given weekend is 0 7901, correct to four decimal places.
45. 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 2
3
(i) Show that the probability that Megan earns one point for a
given weekend is 0 7901, correct to four decimal places.
Let X be the number of matches picked correctly
46. 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 2
3
(i) Show that the probability that Megan earns one point for a
given weekend is 0 7901, correct to four decimal places.
Let X be the number of matches picked correctly
2 3 1 4 0 5
P X 35C3 5C4 5C5
1 2 1 2 1 2
3 3 3 3 3 3
47. 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 2
3
(i) Show that the probability that Megan earns one point for a
given weekend is 0 7901, correct to four decimal places.
Let X be the number of matches picked correctly
2 3 1 4 0 5
P X 35C3 5C4 5C5
1 2 1 2 1 2
3 3 3 3 3 3
5
C3 23 5C4 2 4 5C5 25
35
48. 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 2
3
(i) Show that the probability that Megan earns one point for a
given weekend is 0 7901, correct to four decimal places.
Let X be the number of matches picked correctly
2 3 1 4 0 5
P X 35C3 5C4 5C5
1 2 1 2 1 2
3 3 3 3 3 3
5
C3 23 5C4 2 4 5C5 25
35
0 7901
49. (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.
50. (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
51. (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
PY 1818C18 0 2099 0 7901
0 18
52. (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
PY 1818C18 0 2099 0 7901
0 18
0 01 (to 2 dp)
53. (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
PY 1818C18 0 2099 0 7901
0 18
0 01 (to 2 dp)
(iii) Find the probability that Megan earns at most 16 points
during the eighteen week season. Give your answer correct
to two decimal places.
54. (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
PY 1818C18 0 2099 0 7901
0 18
0 01 (to 2 dp)
(iii) Find the probability that Megan earns at most 16 points
during the eighteen week season. Give your answer correct
to two decimal places.
PY 16 1 PY 17
55. (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
PY 1818C18 0 2099 0 7901
0 18
0 01 (to 2 dp)
(iii) Find the probability that Megan earns at most 16 points
during the eighteen week season. Give your answer correct
to two decimal places.
PY 16 1 PY 17
118C17 0 2099 0 7901 18C18 0 2099 0 7901
1 17 0 18
56. (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
PY 1818C18 0 2099 0 7901
0 18
0 01 (to 2 dp)
(iii) Find the probability that Megan earns at most 16 points
during the eighteen week season. Give your answer correct
to two decimal places.
PY 16 1 PY 17
118C17 0 2099 0 7901 18C18 0 2099 0 7901
1 17 0 18
0 92 (to 2 dp)
57. 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?
58. 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?
P 2 green 0.1 0.1
0.01
59. 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?
P 2 green 0.1 0.1
0.01
(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.
60. 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?
P 2 green 0.1 0.1
0.01
(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.
Let X be the number of people with green eyes
61. 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?
P 2 green 0.1 0.1
0.01
(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.
Let X be the number of people with green eyes
P X 2 C2 0.9 0.1
20 18 2
62. 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?
P 2 green 0.1 0.1
0.01
(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.
Let X be the number of people with green eyes
P X 2 C2 0.9 0.1
20 18 2
0.2851
0.285 (to 3 dp)
63. (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.
64. (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.
P X 2 1 P X 2
65. (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.
P X 2 1 P X 2
1 C2 0 9 0 1 C1 0 9 0 1 C0 0 9 0 1
20 18 2 20 19 1 20 20 0
66. (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.
P X 2 1 P X 2
1 C2 0 9 0 1 C1 0 9 0 1 C0 0 9 0 1
20 18 2 20 19 1 20 20 0
0.3230
0 32 (to 2 dp)
67. (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.
P X 2 1 P X 2
1 C2 0 9 0 1 C1 0 9 0 1 C0 0 9 0 1
20 18 2 20 19 1 20 20 0
0.3230
0 32 (to 2 dp)
Exercise 10J;
1 to 24 even