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Official PYQ · 2011Q34Probability: Random Variables and DistributionsMCQ1 mark

GATE CS 2011 Q34 — Probability: Random Variables and Distributions

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A deck of 55 cards (each carrying a distinct number from 11 to 55) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card?
 

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Answer: A

To find the probability that the first card drawn is exactly one higher than the second card, we can analyze the sample space and the favorable outcomes.

Step 1: Find the total number of outcomes
We are drawing two cards sequentially without replacement from a deck of 55 cards (numbered 11 to 55).

  • Number of choices for the first card (X1X_1) = 55
  • Number of choices for the second card (X2X_2) = 44
Total number of possible outcomes in the sample space SS is: ∣S∣=5×4=20|S| = 5 \times 4 = 20

Step 2: Find the number of favorable outcomes
We want the first card to be one higher than the second card, which means: X1=X2+1X_1 = X_2 + 1 Since the cards are numbered from 11 to 55, the possible pairs (X1,X2)(X_1, X_2) that satisfy this condition are:

  1. (2,1)(2, 1)
  2. (3,2)(3, 2)
  3. (4,3)(4, 3)
  4. (5,4)(5, 4)
So, there are exactly 44 favorable outcomes.

Step 3: Calculate the probability
P(X1=X2+1)=Favorable outcomesTotal outcomes=420=15P(X_1 = X_2 + 1) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{4}{20} = \frac{1}{5}

Therefore, the correct option is A.

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