Probability and odds for Drawing/Picking/Choosing an Ace, a Clubs from a Pack of cards

Problem 2

If a card is drawn from a pack of cards, then find the probability and odds for the card to be (a) an ace and (b) a club.
Ans : (a)
1
13
, 1 : 12, (b)
1
4
, 1 : 3

Solution

Experiment :

Drawing a card from a pack of 52 cards

Total Number of Possible Choices

= Number of ways in which one card can be drawn from the total 52

⇒ n = 52C1
=
52
1
= 52

Let

A : the event of the card drawn being an ace.

B : the event of the card drawn being a club.

For Event A

Favorable
(Aces)
Unfavorable
(Others)
Total
Available 4 48 52
To Choose 1 0 1
Choices 4C148C052C1

Number of Favorable Choices

= Number of ways in which one card can be drawn from the total 4 favorable cards

⇒ mA = 4C1
=
4
1
= 4

Probability of the card drawn being an ace

⇒ 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
=
4
52
=
1
13

Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

⇒ mAc = n − mA
= 52 − 4
= 48

in favor

Odds in Favor of the card drawn being an ace

⇒ Odds in Favor of Event A

= Number of Favorable Choices : Number of Unfavorable Choices

= mA : mAc

= 4 : 48

= 1 : 12

against

Odds against the card drawn being an ace

⇒ Odds against Event A

= Number of Unfavorable Choices : Number of Favorable Choices

= mAc : mA

= 48 : 4

= 12 : 1

For Event B

Favorable
(Clubs)
Others Total
Available 13 39 52
To Choose 1 0 1
Choices 13C139C052C1

Number of Favorable Choices

= Number of ways in which one card can be drawn from the total 13 favorable cards

⇒ mA = 13C1
=
13
1
= 13

Probability of the card drawn being a club

⇒ Probability of occurrence of Event B

=
Number of Favorable Choices for the Event
Total Number of Possible Choices for the Experiment
⇒ P(B) =
mB
n
=
13
52
=
1
4

Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

⇒ mAc = n − mA
= 52 − 13
= 39

in favor

Odds in Favor of the card drawn being a club

⇒ Odds in Favor of Event B

= Number of Favorable Choices : Number of Unfavorable Choices

= mB : mBc

= 13 : 39

= 1 : 3

against

Odds against the card drawn being a club

⇒ Odds against Event B

= Number of Unfavorable Choices : Number of Favorable Choices

= mBc : mB

= 39 : 13

= 3 : 1