A card is drawn from a deck of 52 cards. The outcome is noted, the card is replaced and the deck reshuffled. Another card is then drawn from the deck. i. What is the probability that both the cards are of the same suit? ii. What is the probability that the first card is an ace and the second card is a red queen?

Question

Philoid · Accepted Answer

Given that two cards are drawn from the deck with replacement.

We know that there will four suits in a deck and each suit contains 13 cards namely Spades, Hearts, Diamonds, Clubs.

Also, there will total of 4 aces and 2 red queens and 2 black queens

Let us find the probability required:

⇒ P(D_same) = P(For selecting a card from a single suit out of 52 cards)

⇒

This probability will be same for all the suits.

⇒

⇒ P(D_ace) = P(Drawing an ace)

⇒

⇒ P(D_redQ) = P(Drawing a red queen)

⇒

We need to find the probability of getting:

i. Both cards from same deck

ii. First an ace and second a red queen

⇒ P(S_same) = P(getting both cards from same deck)

We may get two cards any of the four decks. So, each deck’s probability is taken into consideration.

⇒ P(S_same) = P(both cards from spade) + P(both cards from hearts) + P(both cards from diamond) + P(both cards from club)

Since drawing a card is an independent event, their probabilities multiply each other.

⇒ P(S_same) = (P(D_S)P(D_S)) + (P(D_H)P(D_H)) + (P(D_D)P(D_D)) + (P(D_C)P(D_C))

⇒

⇒ P(S_AR) = P(getting an ace first and followed red queen)

Since drawing a card is an independent event, their probabilities multiply each other.

⇒ P(S_AR) = (P(D_A)P(D_redQ))

⇒

∴ The required probabilities are .