Question

14 you randomly shuffle a standard deck of cards with no jokers. this deck has 4 suits (diamonds, spades, hearts, clubs), with 13 cards per suit (numbers 2 through 10, jack, queen, king, ace), which add up to 52 cards in total. after shuffling, you deal two cards from the top. what is the probability that you deal a card in clubs followed by a card in hearts? submit your answer as a simplified fraction or a decimal rounded to 4 decimal places. show your work here enter your answer

Explanation:

Step1: Calculate probability of first - card being a Club

There are 13 clubs in a 52 - card deck. The probability of drawing a club as the first card is the number of clubs divided by the total number of cards. So, $P(\text{Club first})=\frac{13}{52}=\frac{1}{4}$.

Step2: Calculate probability of second - card being a Heart given first is a Club

After drawing a club as the first card, there are 51 cards left in the deck. There are 13 hearts. So the probability of drawing a heart as the second card given that the first card was a club is $P(\text{Heart second}|\text{Club first})=\frac{13}{51}$.

Step3: Calculate the joint probability

By the multiplication rule for conditional probability $P(A\cap B)=P(A)\times P(B|A)$. Here, $A$ is the event of drawing a club first and $B$ is the event of drawing a heart second. So $P(\text{Club then Heart})=\frac{13}{52}\times\frac{13}{51}=\frac{13\times13}{52\times51}=\frac{169}{2652}=\frac{13}{204}\approx0.0637$.

Answer:

$\frac{13}{204}\approx0.0637$