Question 47 of 55intermediate🔧 ApplyNumerical3 marks

4 cards are drawn from a well – shuffled deck of 52 cards. What is the probability of obtaining 3 diamonds and one spade?

Correct Answer

The probability of obtaining 3 diamonds and one spade is 286/20825.

Exercise: Miscellaneous Exercise on Chapter 14 | Q: 2 | (Page Page 22)
For More Understanding

Explanation

This question is from Miscellaneous Exercise on Chapter 14, Question 2. The solution follows the same approach as Example 10 in the textbook, which uses combinations to calculate probabilities when drawing multiple cards. We need to find favorable outcomes (choosing 3 diamonds from 13 and 1 spade from 13) divided by total possible outcomes (choosing 4 cards from 52).

Solution Steps

  1. Step 1: Total number of possible outcomes when 4 cards are drawn from 52 cards = ⁵²C₄ = (52×5152 \times 51×50×4950 \times 49)/(4×34 \times 3×2×12 \times 1) = 270725

  2. Step 2: Number of ways to choose 3 diamonds from 13 diamonds = ¹³C₃ = (13×1213 \times 12×11)/(3×23 \times 2×1) = 286

  3. Step 3: Number of ways to choose 1 spade from 13 spades = ¹³C₁ = 13

  4. Step 4: Number of favorable outcomes = ¹³C₃ × ¹³C₁ = 286×13286 \times 13 = 3718

  5. Step 5: Required probability = 3718/270725 = 286/20825

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