let two cards be dealt sucessively, without replacement, from a standard 52-card deck. find the probability of each event.
a) club dealt second, given a diamond dealt first.
b) the first card is the ace of hearts and the second is black
(a) Given a diamond dealt first (without replacement)
There are 51 cards left, so 13 clubs from a deck of 51 cards:
P(C|D)=13/51.
(b) Ace of hearts followed by a black card.
P(A∪B)=1/52*26/51=1/102
a) To find the probability of a club being dealt second, given that a diamond was dealt first, we need to determine the number of favorable outcomes and the number of possible outcomes.
Number of favorable outcomes: There are 12 clubs left in the deck (after the diamond has been dealt).
Number of possible outcomes: After the diamond has been dealt, there are 51 cards remaining in the deck.
Therefore, the probability of a club being dealt second, given a diamond dealt first, is:
P(Club dealt second | Diamond dealt first) = Number of favorable outcomes / Number of possible outcomes
= 12 / 51
b) To find the probability of the first card being the ace of hearts and the second card being black, we need to determine the number of favorable outcomes and the number of possible outcomes.
Number of favorable outcomes: There is only one ace of hearts in the deck. There are 26 black cards (spades and clubs) remaining after the ace of hearts has been dealt.
Number of possible outcomes: After the ace of hearts has been dealt, there are 51 cards remaining in the deck.
Therefore, the probability of the first card being the ace of hearts and the second card being black is:
P(Ace of Hearts first, Black second) = Number of favorable outcomes / Number of possible outcomes
= 1 / 51
To find the probability of each event, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event.
First, let's calculate the total number of possible outcomes when two cards are dealt successively from a standard 52-card deck:
Total number of possible outcomes = Total number of cards in the deck * Total number of cards remaining after the first draw
= 52 * 51 (since the first card can be any of the 52 cards and the second card can be any of the remaining 51 cards)
a) Club dealt second, given a diamond dealt first:
For this event, we need to find the number of favorable outcomes, which is when a club is dealt second, given that a diamond was dealt first.
Number of favorable outcomes = Number of diamonds in the deck * Number of clubs remaining after the diamond is drawn
= 13 * 12 (as there are 13 diamonds in the deck and 12 clubs remaining after the diamond is drawn)
Therefore, the probability of this event is:
Probability = Number of favorable outcomes / Total number of possible outcomes
= (13 * 12) / (52 * 51)
b) The first card is the Ace of Hearts and the second is black:
For this event, we need to find the number of favorable outcomes, which is when the first card is the Ace of Hearts and the second card is black.
Number of favorable outcomes = 1 (since there is only one Ace of Hearts in the deck) * Number of black cards remaining after the Ace of Hearts is drawn
= 1 * 26 (as there are 26 black cards remaining after the Ace of Hearts is drawn - 26 red cards)
Therefore, the probability of this event is:
Probability = Number of favorable outcomes / Total number of possible outcomes
= (1 * 26) / (52 * 51)
Please note that both probabilities can be simplified further if needed.