Probability of drawing a 10 not of hearts and another hearts card

Problem 4

From a pack of cards, 2 cards are chosen at random. Find the probability of the event of getting a 10 not of hearts and another hearts cards.

Solution

Total number of cards in the pack

= 52

Number of cards drawn

= 2

Experiment :

Drawing 2 cards from the pack of cards

Total Number of Possible Choices

= Number of ways in which 2 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
2!
= 26 × 51
= 1,326

Let

A : the event of drawing a 10 not of hearts and another hearts card

For Event A

10 Not
of Hearts
Hearts Others Total
Available 3 12 37 52
To Choose 1 1 0 2
Choices 3C112C137C052C2

Number of Favorable Choices

= Number of ways in which a 10 not of hearts and another hearts card can be drawn from the total 52

= Number of ways in which a 10 not of hearts can be drawn from the available 3 × Number of ways in which a hearts card can be drawn from the available 12

⇒ mA = 3C1 × 12C1
=
3
1
×
12
1
= 3 × 12
= 36

Probability of choosing a 10 not of hearts and another hearts card

⇒ 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
=
36
1,326
=
6
221

Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

⇒ mAc = n − mA
= 1,326 − 36
= 1,290

in favor

Odds in Favor of choosing a 10 not of hearts and another hearts card

⇒ Odds in Favor of Event A

= Number of Favorable Choices : Number of Unfavorable Choices

= mA : mAc

= 36 : 1,290

= 6 : 215

against

Odds against choosing a 10 not of hearts and another hearts card

⇒ Odds against Event A

= Number of Unfavorable Choices : Number of Favorable Choices

= mAc : mA

= 1,290 : 36

= 215 : 6

Odds (alternative)

Probability of non-occurrence of Event A

⇒ P(Ac) = 1 − P(A)
= 1 −
6
221
=
221 − 6
221
=
215
221

in favor

Odds in Favor of choosing a 10 not of hearts and another hearts card

⇒ Odds in Favor of Event A

= Probability of occurrence of the event : Probability of non-occurrence of the event
=
6
221
:
215
221
= 6 : 215

against

Odds against choosing a 10 not of hearts and another hearts card

⇒ Odds against Event A

= Probability of non-occurrence of the event : Probability of occurrence of the event
=
215
221
:
6
221
= 221 : 6