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Two cards are drawn at random from a pack of 52 cards. What is the probability that either both are black or both are jacks?
- a)62/221
- b)21/312
- c)5/21
- d)55/221
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Two cards are drawn at random from a pack of 52 cards. What is the pro...
There are total 26 cards black, and 4 jacks in which 2 are black jacks
So case 1: both are black
26C2 / 52C2
case 2: both are jack
4C2 / 52C2
Add both cases.
But now 2 black jacks have been added in both cases, so subtracting their prob. :
2C2 / 52C2
So 325/1326 + 6/1326 – 1/1326
So case 1: both are black
26C2 / 52C2
case 2: both are jack
4C2 / 52C2
Add both cases.
But now 2 black jacks have been added in both cases, so subtracting their prob. :
2C2 / 52C2
So 325/1326 + 6/1326 – 1/1326
Most Upvoted Answer
Two cards are drawn at random from a pack of 52 cards. What is the pro...
To find the probability that either both cards are black or both cards are jacks, we need to calculate the probability of each event separately and then subtract the probability of their intersection.
Probability of both cards being black:
There are 26 black cards in a deck of 52 cards. The probability of choosing a black card on the first draw is 26/52. After the first card is drawn, there are 25 black cards left out of the remaining 51 cards. So, the probability of choosing a black card on the second draw, given that the first card was black, is 25/51. Therefore, the probability of both cards being black is:
(26/52) * (25/51) = 25/102
Probability of both cards being jacks:
There are 4 jacks in a deck of 52 cards. The probability of choosing a jack on the first draw is 4/52. After the first card is drawn, there are 3 jacks left out of the remaining 51 cards. So, the probability of choosing a jack on the second draw, given that the first card was a jack, is 3/51. Therefore, the probability of both cards being jacks is:
(4/52) * (3/51) = 1/221
To find the probability of either event occurring, we add the probabilities of each event and subtract the probability of their intersection:
(25/102) + (1/221) - (1/221) = 25/102
Therefore, the probability that either both cards are black or both cards are jacks is 25/102, which is equivalent to 55/221.
Hence, the correct answer is option D, 55/221.
Probability of both cards being black:
There are 26 black cards in a deck of 52 cards. The probability of choosing a black card on the first draw is 26/52. After the first card is drawn, there are 25 black cards left out of the remaining 51 cards. So, the probability of choosing a black card on the second draw, given that the first card was black, is 25/51. Therefore, the probability of both cards being black is:
(26/52) * (25/51) = 25/102
Probability of both cards being jacks:
There are 4 jacks in a deck of 52 cards. The probability of choosing a jack on the first draw is 4/52. After the first card is drawn, there are 3 jacks left out of the remaining 51 cards. So, the probability of choosing a jack on the second draw, given that the first card was a jack, is 3/51. Therefore, the probability of both cards being jacks is:
(4/52) * (3/51) = 1/221
To find the probability of either event occurring, we add the probabilities of each event and subtract the probability of their intersection:
(25/102) + (1/221) - (1/221) = 25/102
Therefore, the probability that either both cards are black or both cards are jacks is 25/102, which is equivalent to 55/221.
Hence, the correct answer is option D, 55/221.
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