Problem: Four cards are drawn from a pack of 52 cards. The probability that out of 4 cards 2 are red and 2 are black is.

Solution:

To find the probability of drawing 2 red cards and 2 black cards out of 4 cards, we need to consider two scenarios:

Scenario 1: Drawing 2 red cards first, followed by 2 black cards.
Scenario 2: Drawing 2 black cards first, followed by 2 red cards.

We will calculate the probability for each scenario and add them together to get the final probability.

Scenario 1: Drawing 2 red cards first, followed by 2 black cards.

In a deck of 52 cards, there are 26 red cards and 26 black cards. The probability of drawing a red card on the first draw is 26/52. After one red card is drawn, there are 25 red cards left out of 51 cards. So, the probability of drawing another red card on the second draw is 25/51.

Now, we need to draw 2 black cards. After drawing 2 red cards, there are 26 black cards left out of 50 cards. The probability of drawing a black card on the third draw is 26/50. After one black card is drawn, there are 25 black cards left out of 49 cards. So, the probability of drawing another black card on the fourth draw is 25/49.

To find the probability of Scenario 1, we multiply the individual probabilities:

Probability of Scenario 1 = (26/52) * (25/51) * (26/50) * (25/49)

Scenario 2: Drawing 2 black cards first, followed by 2 red cards.

Using similar reasoning as above, the probability of Scenario 2 is:

Probability of Scenario 2 = (26/52) * (25/51) * (26/50) * (25/49)

Final Probability:

To find the final probability, we add the probabilities of both scenarios:

Final Probability = Probability of Scenario 1 + Probability of Scenario 2

Final Probability = (26/52) * (25/51) * (26/50) * (25/49) + (26/52) * (25/51) * (26/50) * (25/49)

Final Probability = (26/52) * (25/51) * (26/50) * (25/49) * 2

Simplifying the expression, we get:

Final Probability = (13/52) * (5/17) * (13/25) * (5/49) * 2

Final Probability = (325/833)

Therefore, the probability that out of 4 cards, 2 are red and 2 are black is 325/833, which corresponds to option (a).

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