Two cards are drawn at random from a well shuffled pack of playing cards.
Find the probability that
i. Both are Red
ii. One is Heart and other is Diamond.
iii. Both are of same colour.
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Answered by
0
Answer :-
Total possible outcomes =52
1. Favourable outcome ( both red cards ) = 2
Probability = No. of favourable outcome / Total no. of outcome
= 2/52
= 1/ 26
2. Favourable outcome ( one is heart and other is diamond ) = 2
Probability = 2/52 = 1/26
3. Favourable outcome ( both of same colour ) = 2
Probability = 2/52 = 1/26
Answered by
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Let
A
be the event of picking two queens from the pack of cards, and
B
be the event of picking two red cards.
We want to find (∪)=()+()−(∩)
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
which follows from the principle of inclusion exclusion.
()=
P
(
A
)
=
the probability of picking two queens. Since there are 4 queens, all of which are equally likely to be chosen, and we can choose 2 queens from 4 queens in (42)
(
4
2
)
ways, out of a total (522)
(
52
2
)
ways of choosing any two cards from a deck. So ()=(42)(522)
P
(
A
)
=
(
4
2
)
(
52
2
)
.
()=
P
(
B
)
=
the probability of picking two red cards. Since there are 13×2=26
13
×
2
=
26
red cards, ()=(262)(522)
P
(
B
)
=
(
26
2
)
(
52
2
)
.
(∩)=
P
(
A
∩
B
)
=
the probability of picking up two red queens. Since there are only two red queens, this can be done in only one way, so we have here, (∩)=1(522)
P
(
A
∩
B
)
=
1
(
52
2
)
.
So, the required probability, can be calculated from the formula given above since we want to find the probability of the event that we have two red cards, or two queens, that is, of the event ∪
A
∪
B
.
A
be the event of picking two queens from the pack of cards, and
B
be the event of picking two red cards.
We want to find (∪)=()+()−(∩)
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
which follows from the principle of inclusion exclusion.
()=
P
(
A
)
=
the probability of picking two queens. Since there are 4 queens, all of which are equally likely to be chosen, and we can choose 2 queens from 4 queens in (42)
(
4
2
)
ways, out of a total (522)
(
52
2
)
ways of choosing any two cards from a deck. So ()=(42)(522)
P
(
A
)
=
(
4
2
)
(
52
2
)
.
()=
P
(
B
)
=
the probability of picking two red cards. Since there are 13×2=26
13
×
2
=
26
red cards, ()=(262)(522)
P
(
B
)
=
(
26
2
)
(
52
2
)
.
(∩)=
P
(
A
∩
B
)
=
the probability of picking up two red queens. Since there are only two red queens, this can be done in only one way, so we have here, (∩)=1(522)
P
(
A
∩
B
)
=
1
(
52
2
)
.
So, the required probability, can be calculated from the formula given above since we want to find the probability of the event that we have two red cards, or two queens, that is, of the event ∪
A
∪
B
.
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