Question

conditional probability
two cards are drawn from a deck of 52 cards. determine the following probabilites. write your answers
as simplified fractions.
note: aces are not considered number cards or face cards.
deck of cards
what is the probability the second card is a diamond given that the first card is diamond?
what is the probability the second card is a club given that the first card is spade?
what is the probability the second card is a jack given that the first card is a 5 of clubs?
what is the probability the second card is a number card given that the first card is a 5 of hearts?

Explanation:

Step1: Calculate the probability for the first question

There are 13 diamonds in a deck. If the first card is a diamond, then there are 12 diamonds left and 51 cards left.
The probability is \(P=\frac{12}{51}=\frac{4}{17}\)

Step2: Calculate the probability for the second question

There are 13 clubs in a deck. The first - card is a spade (which has no effect on the number of clubs remaining). There are still 13 clubs and 51 cards left.
The probability is \(P = \frac{13}{51}\)

Step3: Calculate the probability for the third question

There are 4 Jacks in a deck. The first card is a 5 of clubs (which has no effect on the number of Jacks remaining). There are still 4 Jacks and 51 cards left.
The probability is \(P=\frac{4}{51}\)

Step4: Calculate the probability for the fourth question

There are 36 number cards (\(9\) number cards per suit, \(4\) suits: \(9\times4 = 36\)). The first card is a 5 of hearts (a number card). So there are \(35\) number cards left and 51 cards left.
The probability is \(P=\frac{35}{51}\)

Answer:

1. \(\frac{4}{17}\)
2. \(\frac{13}{51}\)
3. \(\frac{4}{51}\)
4. \(\frac{35}{51}\)