Transcript Revision Sheet 1

Revision Sheet 1
Question 1a
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3
find the probability of the following events:
a. P(AÈ B)
Question 1a
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3
find the probability of the following events:
a.
P(A Ç B) = P(A)´ P(B) (Independent)
0.3 = 0.6 ´ P(B) Þ P(B) = 0.5
P(A È B) = P(A)+ P(B) - P(A Ç B)
= 0.6 + 0.5 - 0.3 = 0.8
Question 1b
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3
find the probability of the following events:
b.
P(A / B) =
Question 1b
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3
find the probability of the following events:
b.
P(A)´ P(B)
P(A / B) =
= P(A) = 0.6 (Independent)
P(B)
Question 1c
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3
find the probability of the following events:
c.
P(B / A) =
Question 1c
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3
find the probability of the following events:
c.
P(A)´ P(B)
P(B / A) =
= P(B) = 0.5 (Independent)
P(A)
Question 1d
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3
find the probability of the following events:
d.
P(A / B¢)
Question 1d
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3
find the probability of the following events:
d.
P(AÇ B¢) 0.3
P(A / B¢) =
=
= 0.6 (= P(A))
P( B¢)
0.5
Question using a Venn diagram
Two independent events A and B are such that
p(A) = 0.6 and p(AÇ B) = 0.3 p(B) =0.5
A
0.3
B
0.3
0.2
0.2
AÈ B = 0.8
A
0.3
B
0.3
0.2
0.2
0.3
A/B=
= 0.6
0.5
A
0.3
B
0.3
0.2
0.2
0.3
B/A=
= 0.5
0.6
A
0.3
B
0.3
0.2
0.2
0.3
A / B¢ =
= 0.6
0.5
A
0.3
B
0.3
0.2
0.2
Question 2
A and B are two events such that p ( A) = 0.3
p ( B) = 0.5 and p ( AÈ B) = 0.55
find the probability of the following events:
a. A / B
b. B / A
c. A / B¢
d. A¢ / B¢
Question 2
A and B are two events such that p ( A) = 0.3
p ( B) = 0.5 and p ( AÈ B) = 0.55
find the probability of the following events:
a. A / B
b. B / A
c. A / B¢
d. A¢ / B¢
The question does not say the
events are independent so we
calculate p ( AÇ B) first.
Question 2
A and B are two events such that p ( A) = 0.3
p ( B) = 0.5 and p ( AÈ B) = 0.55
find the probability of the following events:
a. A / B
b. B / A
c. A / B¢
d. A¢ / B¢
p ( A È B) = p ( A) + p ( B) - p ( A Ç B)
0.55 = 0.3+ 0.5 - p ( A Ç B)
p ( A Ç B) = 0.25
Question 2
p ( A) = 0.3
p ( B) = 0.5 p ( AÈ B) = 0.55 p ( AÇ B) = 0.25
p ( A Ç B ) 0.25
a. A / B =
=
= 0.5
p ( B)
0.5
b.
p ( A Ç B) 0.25 5
B/A=
=
=
p ( A)
0.3 6
Question 2- easier with a Venn
diagram
p ( A) = 0.3
p ( B) = 0.5 p ( AÈ B) = 0.55 p ( AÇ B) = 0.25
c. A / B¢
d. A¢ / B¢
0.05
c. A / B¢ =
= 0.1
0.5
A
0.05
B
0.25
0.45
0.25
0.05
c. A / B¢ =
= 0.1
0.5
0.45
d. A¢ / B¢ =
= 0.9
0.5
A
B
0.45
0.05
0.25
0.25
Question 3
Urn A contains 9 cubes of which 4 are red. Urn B
contains 5 cubes of which 2 are red. A cube is
drawn at random and in succession from each urn.
a. Draw a tree diagram representing this process.
b. Find the probability that both cubes are red.
c. Find the probability that only 1 cube is red.
d. If only 1 cube is red, find the probability that it
came from urn A.
Draw a tree diagram representing this process.
2/5
R
R
3/5
4/9
R’
2/5
5/9
R
R’
3/5
R’
b. Find the probability that both cubes are red.
2/5
R
4 2 8
´ =
9 5 45
R
3/5
4/9
R’
2/5
5/9
R
R’
3/5
R’
c. Find the probability that only 1 cube is red.
2/5
R
4 3 5 2 22
´ + ´ =
9 5 9 5 45
R
3/5
4/9
R’
2/5
5/9
R
R’
3/5
R’
If only 1 cube is red, find the probability that it
R
came from urn A.
2/5
R
3/5
4/9
R’
2/5
5/9
R’
3/5
R
4 3
´
12 6
9 5
=
=
4 3 5 2 22 11
´ + ´
9 5 9 5
R’
Question 4
A box contains 5 red, 3 black and 2 white cubes. A
cube is randomly drawn and its colour noted. The
cube is then replaced, together with 2 more of the
same colour. A second cube is then drawn.
a. Find the probability that the first cube selected
is red.
b. Find the probability that the second cube is
black.
c. Given that the first cube selected was red, what
is the probability that the second cube selected is
black?
Question 4
A box contains 5 red, 3 black and 2 white cubes.
A cube is randomly drawn and its colour noted.
The cube is then replaced, together with 2 more
of the same colour. A second cube is then
drawn.
a. Find the probability that the first cube
selected is red.
5/10
Question 4
A box contains 5 red, 3 black and 2 white cubes.
A cube is randomly drawn and its colour noted.
The cube is then replaced, together with 2 more
of the same colour. A second cube is then
drawn.
b. Find the probability that the second cube is
black. Combinations are
RB,
BB,
WB
5 3 3 5 2 3 3
´ + ´ + ´ =
10 12 10 12 10 12 10
Question 4
A box contains 5 red, 3 black and 2 white cubes.
A cube is randomly drawn and its colour noted.
The cube is then replaced, together with 2 more
of the same colour. A second cube is then
drawn.
c. Given that the first cube selected was red,
what is the probability that the second cube
selected is black?
3/12
Question 5
A fair coin, a double-headed coin and a doubletailed coin are placed in a bag. A coin is
randomly selected. The coin is then tossed.
a. Draw a tree diagram showing the possible
outcomes.
b. Find the probability that the coin lands with
a tail showing uppermost.
c. In fact, the coin falls “heads”, find the
probability that it is the “double-headed” coin.
Draw a tree diagram
showing the possible
outcomes.
1/2
H
fair
1/3
1/2
1/3
head
1
0
1/3
tail
0
1
T
H
T
H
T
b. Find the probability
that the coin lands with a
tail showing uppermost. 1/2
H
fair
1/3
1/2
1/3
head
0
1/3
tail
1 1 1 1
´ + =
3 2 3 2
1
0
1
T
H
T
H
T
In fact, the coin falls “heads”,
find the probability that it is
the “double-headed” coin.1/2
H
fair
1/3
1/2
1/3
head
0
1/3
1
2
3
=
1 1 1 3
´ +
3 2 3
1
tail
0
1
T
H
T
H
T
Question 6
Two unbiased coins are tossed together. Find
the probability that they both display heads
given that at least one is showing a head.
Question 6
Two unbiased coins are tossed together. Find
the probability that they both display heads
given that at least one is showing a head.
1/3