Post on 22-Nov-2014
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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