If 7 cards are drawn at random. From a pack of 52 cards then find the probability that 3 of them are red and 4 are black.

Total number of cards in the pack

= 52

Number of cards drawn

= 7

Experiment :

Drawing 7 cards from the pack of cards

Total Number of Possible Choices

= Number of ways in which 7 cards can be drawn from the 52 cards

⇒ n	=	52C7 When simplification leads to large calculations leaving the term as it is may help. Simplify when needed in the steps while deriving the required answer.
	=	52 × 51 × ... 7 terms 7!
	=	52 × 51 × 50 × 49 × 48 × 47 × 46 7 × 6 × 5 × 4 × 3 × 2 × 1
	=	52 × 17 × 10 × 7 × 47 × 46

Let

A : the event of drawing 3 red 4 black cards

For Event A

	Red	Black	Total
Available	26	26	52
To Choose	3	4	7
Choices	26C3	26C4	52C7

Number of Favorable Choices

= Number of ways in which 3 red and 4 black cards can be drawn from the total 52

= Number of ways in which 3 red cards can be drawn from the available 26 × Number of ways in which 4 black cards can be drawn from the available 26

⇒ mA	=	26C3 × 26C4
	=	26 × 25 × 24 3 × 2 × 1 × 26 × 25 × 24 × 23 4 × 3 × 2 × 1
	=	(26 × 25 × 4) × (26 × 25 × 23)

Probability of drawing 3 red 4 black cards

⇒ Probability of occurrence of Event A

=	Number of Favorable Choices for the Event Total Number of Possible Choices for the Experiment

⇒ P(A)	=	mA n
	=	26C3 × 26C4 52C7
	=	26 × 25 × 24 3 × 2 × 1 × 26 × 25 × 24 × 23 4! × 7 × 6 × 5 × 4! 52 × 51 × 50 × 49 × 48 × 47 × 46
	=	5 × 23 × 25 17 × 7 × 47
	=	2,875 5,593

Odds

Probability of non-occurrence of Event A

⇒ P(Ac)	=	1 − P(A)
	=	1 − 1 273
	=	5,593 − 2,875 5,593
	=	2,718 5,593

in favor

Odds in Favor of drawing 3 red 4 black cards

⇒ Odds in Favor of Event A

=	Probability of occurrence of the event : Probability of non-occurrence of the event
=	2,875 5,593 : 2,718 5,593
=	2,875 : 2,718

against

Odds against drawing 3 red 4 black cards

⇒ Odds against Event A

=	Probability of non-occurrence of the event : Probability of occurrence of the event
=	2,718 5,593 : 2,875 5,593
=	2,718 : 2,875