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From a pack of 52 cards, 2 cards are drawn at random. What is the probability that either both are red or both are kings?
- a)55/221
- b)52/225
- c)44/221
- d)48/221
- e)None of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
From a pack of 52 cards, 2 cards are drawn at random. What is the prob...
Prob. of both red = 26C2 / 52C2
Prob. of both kings = 4C2 / 52C2
Since there are also cads which are both red and king, so we will subtract there prob.
There are 2 red cards which are kings
Prob. of both red and king = 2C2 / 52C2
So required prob. = 26C2 / 52C2 + 4C2 / 52C2 – 2C2 / 52C2
Prob. of both kings = 4C2 / 52C2
Since there are also cads which are both red and king, so we will subtract there prob.
There are 2 red cards which are kings
Prob. of both red and king = 2C2 / 52C2
So required prob. = 26C2 / 52C2 + 4C2 / 52C2 – 2C2 / 52C2
Most Upvoted Answer
From a pack of 52 cards, 2 cards are drawn at random. What is the prob...
Understanding the Problem
To find the probability that either both cards drawn are red or both are kings from a standard deck of 52 cards, we will use the principle of inclusion-exclusion.
Step 1: Calculate the Probability of Drawing Two Red Cards
- There are 26 red cards in a deck (13 hearts and 13 diamonds).
- The number of ways to choose 2 red cards is given by the combination formula:
\[
C(26, 2) = \frac{26!}{2!(26-2)!} = \frac{26 \times 25}{2} = 325
\]
- The total number of ways to choose any 2 cards from 52 is:
\[
C(52, 2) = \frac{52!}{2!(52-2)!} = \frac{52 \times 51}{2} = 1326
\]
- Thus, the probability of drawing 2 red cards is:
\[
P(\text{2 red}) = \frac{325}{1326}
\]
Step 2: Calculate the Probability of Drawing Two Kings
- There are 4 kings in the deck.
- The number of ways to choose 2 kings is:
\[
C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2} = 6
\]
- Therefore, the probability of drawing 2 kings is:
\[
P(\text{2 kings}) = \frac{6}{1326}
\]
Step 3: Calculate the Probability of Both Events Occurring
- Since there are no red kings, the events are mutually exclusive, and we can simply add the probabilities:
\[
P(\text{2 red or 2 kings}) = P(\text{2 red}) + P(\text{2 kings}) = \frac{325}{1326} + \frac{6}{1326} = \frac{331}{1326}
\]
Step 4: Simplifying the Probability
- Now, simplify \(\frac{331}{1326}\):
\[
P = \frac{331}{1326} = \frac{55}{221}
\]
Thus, the correct answer is option 'A': \(\frac{55}{221}\).
To find the probability that either both cards drawn are red or both are kings from a standard deck of 52 cards, we will use the principle of inclusion-exclusion.
Step 1: Calculate the Probability of Drawing Two Red Cards
- There are 26 red cards in a deck (13 hearts and 13 diamonds).
- The number of ways to choose 2 red cards is given by the combination formula:
\[
C(26, 2) = \frac{26!}{2!(26-2)!} = \frac{26 \times 25}{2} = 325
\]
- The total number of ways to choose any 2 cards from 52 is:
\[
C(52, 2) = \frac{52!}{2!(52-2)!} = \frac{52 \times 51}{2} = 1326
\]
- Thus, the probability of drawing 2 red cards is:
\[
P(\text{2 red}) = \frac{325}{1326}
\]
Step 2: Calculate the Probability of Drawing Two Kings
- There are 4 kings in the deck.
- The number of ways to choose 2 kings is:
\[
C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2} = 6
\]
- Therefore, the probability of drawing 2 kings is:
\[
P(\text{2 kings}) = \frac{6}{1326}
\]
Step 3: Calculate the Probability of Both Events Occurring
- Since there are no red kings, the events are mutually exclusive, and we can simply add the probabilities:
\[
P(\text{2 red or 2 kings}) = P(\text{2 red}) + P(\text{2 kings}) = \frac{325}{1326} + \frac{6}{1326} = \frac{331}{1326}
\]
Step 4: Simplifying the Probability
- Now, simplify \(\frac{331}{1326}\):
\[
P = \frac{331}{1326} = \frac{55}{221}
\]
Thus, the correct answer is option 'A': \(\frac{55}{221}\).
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