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Math Problems
Precalculus
Identify independent events
2
2
2
. For two events
A
A
A
and
B
,
P
(
A
)
=
0.4
,
P
(
B
)
=
0.2
B, P(A)=0.4, P(B)=0.2
B
,
P
(
A
)
=
0.4
,
P
(
B
)
=
0.2
and
P
(
A
∣
B
)
=
0.6
P(A \mid B)=0.6
P
(
A
∣
B
)
=
0.6
\newline
1
1
1
\newline
(a) Find
P
(
A
∩
B
)
P(A \cap B)
P
(
A
∩
B
)
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8
8
8
[N
2010
2010
2010
/II/
7
7
7
modified] For events
A
A
A
and
B
B
B
, it is given that
P
(
A
)
=
0.7
,
P
(
B
)
=
0.6
\mathrm{P}(A)=0.7, \mathrm{P}(B)=0.6
P
(
A
)
=
0.7
,
P
(
B
)
=
0.6
and
P
(
A
∣
B
′
)
=
0.8
P\left(A \mid B^{\prime}\right)=0.8
P
(
A
∣
B
′
)
=
0.8
. Find
\newline
(i)
P
(
A
∩
B
′
)
\mathrm{P}\left(A \cap B^{\prime}\right)
P
(
A
∩
B
′
)
,
\newline
(ii)
P
(
A
∪
B
)
\mathrm{P}(A \cup B)
P
(
A
∪
B
)
,
\newline
(iii)
P
(
B
′
∣
A
)
\mathrm{P}\left(B^{\prime} \mid A\right)
P
(
B
′
∣
A
)
.
\newline
For a third event
C
C
C
, it is given that
P
(
C
)
=
0.5
\mathrm{P}(C)=0.5
P
(
C
)
=
0.5
and that
A
A
A
and
C
C
C
are independent.
\newline
(iv) Find
B
B
B
1
1
1
\newline
(v) Hence state an inequality satisfied by
B
B
B
2
2
2
in the form
B
B
B
3
3
3
, where
B
B
B
4
4
4
and
B
B
B
5
5
5
are constants to be determined.
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In an experiment, the probability that event
A
A
A
occurs is
5
9
\frac{5}{9}
9
5
, the probability that event
B
B
B
occurs is
8
9
\frac{8}{9}
9
8
, and the probability that events
A
A
A
and
B
B
B
both occur is
40
81
\frac{40}{81}
81
40
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
5
9
\frac{5}{9}
9
5
, the probability that event
B
B
B
occurs is
3
5
\frac{3}{5}
5
3
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
6
\frac{1}{6}
6
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
3
4
\frac{3}{4}
4
3
, the probability that event
B
B
B
occurs is
1
3
\frac{1}{3}
3
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
8
\frac{1}{8}
8
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
7
\frac{2}{7}
7
2
, the probability that event
B
B
B
occurs is
5
6
\frac{5}{6}
6
5
, and the probability that events
A
A
A
and
B
B
B
both occur is
5
21
\frac{5}{21}
21
5
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
4
5
\frac{4}{5}
5
4
, the probability that event
B
B
B
occurs is
4
7
\frac{4}{7}
7
4
, and the probability that events
A
A
A
and
B
B
B
both occur is
16
35
\frac{16}{35}
35
16
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
3
7
\frac{3}{7}
7
3
, the probability that event
B
B
B
occurs is
2
9
\frac{2}{9}
9
2
, and the probability that events
A
A
A
and
B
B
B
both occur is
2
21
\frac{2}{21}
21
2
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
3
8
\frac{3}{8}
8
3
, the probability that event
B
B
B
occurs is
3
8
\frac{3}{8}
8
3
, and the probability that events
A
A
A
and
B
B
B
both occur is
9
64
\frac{9}{64}
64
9
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
5
7
\frac{5}{7}
7
5
, the probability that event
B
B
B
occurs is
3
4
\frac{3}{4}
4
3
, and the probability that events
A
A
A
and
B
B
B
both occur is
15
28
\frac{15}{28}
28
15
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
7
\frac{1}{7}
7
1
, the probability that event
B
B
B
occurs is
8
9
\frac{8}{9}
9
8
, and the probability that events
A
A
A
and
B
B
B
both occur is
8
63
\frac{8}{63}
63
8
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
3
\frac{2}{3}
3
2
, the probability that event
B
B
B
occurs is
1
2
\frac{1}{2}
2
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
6
\frac{1}{6}
6
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
6
7
\frac{6}{7}
7
6
, the probability that event
B
B
B
occurs is
1
7
\frac{1}{7}
7
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
6
49
\frac{6}{49}
49
6
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
7
\frac{2}{7}
7
2
, the probability that event
B
B
B
occurs is
3
7
\frac{3}{7}
7
3
, and the probability that events
A
A
A
and
B
B
B
both occur is
6
49
\frac{6}{49}
49
6
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
7
\frac{2}{7}
7
2
, the probability that event
B
B
B
occurs is
5
7
\frac{5}{7}
7
5
, and the probability that events
A
A
A
and
B
B
B
both occur is
10
49
\frac{10}{49}
49
10
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
3
8
\frac{3}{8}
8
3
, the probability that event
B
B
B
occurs is
2
3
\frac{2}{3}
3
2
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
8
\frac{1}{8}
8
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
2
\frac{1}{2}
2
1
, the probability that event
B
B
B
occurs is
4
9
\frac{4}{9}
9
4
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
9
\frac{1}{9}
9
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
3
\frac{1}{3}
3
1
, the probability that event
B
B
B
occurs is
1
8
\frac{1}{8}
8
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
24
\frac{1}{24}
24
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
8
9
\frac{8}{9}
9
8
, the probability that event
B
B
B
occurs is
5
6
\frac{5}{6}
6
5
, and the probability that events
A
A
A
and
B
B
B
both occur is
20
27
\frac{20}{27}
27
20
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
9
\frac{1}{9}
9
1
, the probability that event
B
B
B
occurs is
1
9
\frac{1}{9}
9
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
81
\frac{1}{81}
81
1
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
5
\frac{2}{5}
5
2
, the probability that event
B
B
B
occurs is
7
8
\frac{7}{8}
8
7
, and the probability that events
A
A
A
and
B
B
B
both occur is
7
20
\frac{7}{20}
20
7
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
2
\frac{1}{2}
2
1
, the probability that event
B
B
B
occurs is
1
2
\frac{1}{2}
2
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
8
\frac{1}{8}
8
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
3
5
\frac{3}{5}
5
3
, the probability that event
B
B
B
occurs is
5
9
\frac{5}{9}
9
5
, and the probability that events
A
A
A
and
B
B
B
both occur is
2
9
\frac{2}{9}
9
2
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
2
\frac{1}{2}
2
1
, the probability that event
B
B
B
occurs is
2
3
\frac{2}{3}
3
2
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
6
\frac{1}{6}
6
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
8
\frac{1}{8}
8
1
, the probability that event
B
B
B
occurs is
2
7
\frac{2}{7}
7
2
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
28
\frac{1}{28}
28
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
7
\frac{1}{7}
7
1
, the probability that event
B
B
B
occurs is
5
8
\frac{5}{8}
8
5
, and the probability that events
A
A
A
and
B
B
B
both occur is
5
56
\frac{5}{56}
56
5
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
5
9
\frac{5}{9}
9
5
, the probability that event
B
B
B
occurs is
7
9
\frac{7}{9}
9
7
, and the probability that events
A
A
A
and
B
B
B
both occur is
35
81
\frac{35}{81}
81
35
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
5
8
\frac{5}{8}
8
5
, the probability that event
B
B
B
occurs is
2
5
\frac{2}{5}
5
2
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
8
\frac{1}{8}
8
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
4
7
\frac{4}{7}
7
4
, the probability that event
B
B
B
occurs is
1
2
\frac{1}{2}
2
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
7
\frac{1}{7}
7
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
2
\frac{1}{2}
2
1
, the probability that event
B
B
B
occurs is
7
8
\frac{7}{8}
8
7
, and the probability that events
A
A
A
and
B
B
B
both occur is
7
16
\frac{7}{16}
16
7
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
6
\frac{1}{6}
6
1
, the probability that event
B
B
B
occurs is
1
2
\frac{1}{2}
2
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
12
\frac{1}{12}
12
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
3
\frac{2}{3}
3
2
, the probability that event
B
B
B
occurs is
3
8
\frac{3}{8}
8
3
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
8
\frac{1}{8}
8
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
7
\frac{1}{7}
7
1
, the probability that event
B
B
B
occurs is
1
9
\frac{1}{9}
9
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
63
\frac{1}{63}
63
1
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
5
\frac{1}{5}
5
1
, the probability that event
B
B
B
occurs is
1
7
\frac{1}{7}
7
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
35
\frac{1}{35}
35
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
7
\frac{2}{7}
7
2
, the probability that event
B
B
B
occurs is
7
9
\frac{7}{9}
9
7
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
9
\frac{1}{9}
9
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
3
\frac{1}{3}
3
1
, the probability that event
B
B
B
occurs is
3
4
\frac{3}{4}
4
3
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
8
\frac{1}{8}
8
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
8
\frac{1}{8}
8
1
, the probability that event
B
B
B
occurs is
5
6
\frac{5}{6}
6
5
, and the probability that events
A
A
A
and
B
B
B
both occur is
5
48
\frac{5}{48}
48
5
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
9
\frac{1}{9}
9
1
, the probability that event
B
B
B
occurs is
3
7
\frac{3}{7}
7
3
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
21
\frac{1}{21}
21
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
3
5
\frac{3}{5}
5
3
, the probability that event
B
B
B
occurs is
7
8
\frac{7}{8}
8
7
, and the probability that events
A
A
A
and
B
B
B
both occur is
21
40
\frac{21}{40}
40
21
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
3
\frac{2}{3}
3
2
, the probability that event
B
B
B
occurs is
2
5
\frac{2}{5}
5
2
, and the probability that events
A
A
A
and
B
B
B
both occur is
4
15
\frac{4}{15}
15
4
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
5
\frac{1}{5}
5
1
, the probability that event
B
B
B
occurs is
1
4
\frac{1}{4}
4
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
20
\frac{1}{20}
20
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
5
\frac{1}{5}
5
1
, the probability that event
B
B
B
occurs is
7
9
\frac{7}{9}
9
7
, and the probability that events
A
A
A
and
B
B
B
both occur is
7
45
\frac{7}{45}
45
7
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
3
8
\frac{3}{8}
8
3
, the probability that event
B
B
B
occurs is
1
7
\frac{1}{7}
7
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
3
56
\frac{3}{56}
56
3
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
7
8
\frac{7}{8}
8
7
, the probability that event
B
B
B
occurs is
5
6
\frac{5}{6}
6
5
, and the probability that events
A
A
A
and
B
B
B
both occur is
35
48
\frac{35}{48}
48
35
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
2
5
\frac{2}{5}
5
2
, the probability that event
B
B
B
occurs is
5
8
\frac{5}{8}
8
5
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
8
\frac{1}{8}
8
1
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
1
7
\frac{1}{7}
7
1
, the probability that event
B
B
B
occurs is
3
4
\frac{3}{4}
4
3
, and the probability that events
A
A
A
and
B
B
B
both occur is
3
28
\frac{3}{28}
28
3
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
3
8
\frac{3}{8}
8
3
, the probability that event
B
B
B
occurs is
2
7
\frac{2}{7}
7
2
, and the probability that events
A
A
A
and
B
B
B
both occur is
3
28
\frac{3}{28}
28
3
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
8
9
\frac{8}{9}
9
8
, the probability that event
B
B
B
occurs is
4
7
\frac{4}{7}
7
4
, and the probability that events
A
A
A
and
B
B
B
both occur is
32
63
\frac{32}{63}
63
32
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
4
7
\frac{4}{7}
7
4
, the probability that event
B
B
B
occurs is
1
8
\frac{1}{8}
8
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
1
14
\frac{1}{14}
14
1
. Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A)yes
\newline
(B)no
Get tutor help
In an experiment, the probability that event
A
A
A
occurs is
5
6
\frac{5}{6}
6
5
, the probability that event
B
B
B
occurs is
1
3
\frac{1}{3}
3
1
, and the probability that events
A
A
A
and
B
B
B
both occur is
5
18
\frac{5}{18}
18
5
.
\newline
Are
A
A
A
and
B
B
B
independent events?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
Get tutor help
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