12x1 t08 06 binomial probability (2011)
TRANSCRIPT
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