Question

assume that 2 cards are drawn from a standard 52 - card deck. find the following probabilities.
a) assume the cards are drawn without replacement. find the probability of drawing 2 red cards.
b) assume the cards are drawn with replacement. find the probability of drawing 2 red cards.
a. the probability of drawing 2 red cards without replacement is
(simplify your answer.)
b. the probability of drawing 2 red cards with replacement is
(simplify your answer.)

Explanation:

Step1: Calculate probability without replacement

In a standard 52 - card deck, there are 26 red cards.
The probability of drawing the first red card is \(\frac{26}{52}\).
After drawing one red card (without replacement), there are 25 red cards left and 51 cards in total.
The probability of drawing the second red card is \(\frac{25}{51}\).
By the multiplication rule for dependent events \(P(A\cap B)=P(A)\times P(B|A)\), the probability of drawing 2 red cards without replacement is \(\frac{26}{52}\times\frac{25}{51}=\frac{1}{2}\times\frac{25}{51}=\frac{25}{102}\).

Step2: Calculate probability with replacement

The probability of drawing a red card on the first draw is \(\frac{26}{52}=\frac{1}{2}\).
Since the card is replaced, the probability of drawing a red card on the second draw is also \(\frac{26}{52}=\frac{1}{2}\).
By the multiplication rule for independent events \(P(A\cap B) = P(A)\times P(B)\), the probability of drawing 2 red cards with replacement is \(\frac{26}{52}\times\frac{26}{52}=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\).

Answer:

a. \(\frac{25}{102}\)
b. \(\frac{1}{4}\)