A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability that the lost card was a diamond:a)144/169b)11/50c)39/50d)11/24Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Let D – Lost card is diamond
D – Lost card is not diamond.

When one diamond card is lost, the number of diamond cards are 12 out of 51.
No. of ways of choosing 2 from 12 cards ¹²C₂ .
Similarity, choosing 2 diamond cards out of 51 are ⁵¹C₂.

probability of getting two cards, when one diamond is lost

When the lost and is not diamond, number of diamond cards are 13 out of 51.
No. of ways of choosing 2 cards when one card is lost which is not diamond =

Answer

Answer

Problem:
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability that the lost card was a diamond.

Solution:
To solve this problem, we can use Bayes' theorem. Bayes' theorem gives us a way to update the probability of an event based on new information.

Step 1: Identify the events:
Let's define the following events:
A: The lost card is a diamond.
B: The two cards drawn are both diamonds.

Step 2: Determine the probabilities:
We need to find P(A|B), the probability that the lost card is a diamond given that the two cards drawn are both diamonds. Bayes' theorem states:

P(A|B) = (P(B|A) * P(A)) / P(B)

We need to find P(B|A), P(A), and P(B) to calculate P(A|B).

Step 3: Calculate P(B|A):
If the lost card is a diamond, then there are 12 diamonds left in the deck. The probability of drawing a diamond from a deck of 51 cards is 12/51. Since we are drawing two diamonds, the probability of drawing two diamonds given that the lost card is a diamond is:

P(B|A) = (12/51) * (11/50)

Step 4: Calculate P(A):
The probability of the lost card being a diamond is P(A) = 13/52, since there are 13 diamonds in a deck of 52 cards.

Step 5: Calculate P(B):
To calculate P(B), we need to consider two cases:
1. The lost card is a diamond.
2. The lost card is not a diamond.

Case 1: The lost card is a diamond.
As calculated earlier, the probability of drawing two diamonds given that the lost card is a diamond is (12/51) * (11/50). Since there are 13 diamonds in a deck of 52 cards, the probability of this case is:

P(B|A) * P(A) = (12/51) * (11/50) * (13/52)

Case 2: The lost card is not a diamond.
If the lost card is not a diamond, then there are 13 diamonds left in the deck. The probability of drawing two diamonds from a deck of 51 cards is:

P(B|A') * P(A') = (13/51) * (12/50) * (39/52)

Where A' represents the event that the lost card is not a diamond.

The total probability of drawing two diamonds is the sum of the probabilities from both cases:

P(B) = P(B|A) * P(A) + P(B|A') * P(A')

Step 6: Calculate P(A|B):
Finally, we can substitute the calculated values into Bayes' theorem to find P(A|B):

P(A|B) = (P(B|A) * P(A)) / P(B)

Plugging in the values:

P(A|B) = [(12/51) * (11/50) * (13/52)]

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