From a pack of 52 playing cards, face cards and tens are removed and kept aside then a card is drawn at random from the remaining cards. IfA : The event that the card drawn is an aceH : The event that the card drawn is a heartS : The event that the card drawn is a spadethen which of the following holds ?

Engineering

Mathematics

Conditional Probability

Question

From a pack of 52 playing cards, face cards and tens are removed and kept aside then a card is drawn at random from the remaining cards. If
A : The event that the card drawn is an ace
H : The event that the card drawn is a heart
S : The event that the card drawn is a spade
then which of the following holds ?

9P(A) = 4P(H)

P(S) = 4P(A∩H)

3P(H) = 3P(A∪S)

None of these

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Given from a pack of 52 cards, face cards and tens are removed.
⇒ Total no. of remaining cards = 52 − (12 + 4) = 36 cards
A : The event that the card drawn is an ace, $P (A) = \frac{4}{36} = \frac{1}{9}$
H : The event that the card drawn is an Heart, $P (H) = \frac{9}{36} = \frac{1}{4}$
S : The event that the card drawn is a Spade, $P (S) = \frac{9}{36} = \frac{1}{4}$
and $P (A \cap S) = \frac{1}{36}$ (Since only 1 card exist which is spade and ace)
P(A∪S) = P(A) + P(S) − P(A∩S) $= \frac{1}{9} + \frac{1}{4} - \frac{1}{36}$ (since $P (A \cap S) = \frac{1}{36}$ )

$= \frac{1}{3}$

⇒ 9P(A) = 4P(H) = 3P(A∪S) = 1 and 9P(A∩S) = P(H)

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