Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A person draws a card from a pack of 52 playing cards, replaces it and shuffles the pack. He continues doing this until be draws a spade, the chance that he will fail in the first two draws is (A) \(\frac{1}{16}\) (B) \(\frac{9}{16}\) (C) \(\frac{9}{64}\) (D) \(\frac{1}{64}\)

Step-by-Step Solution

Answer

(B) \(\frac{9}{16}\)

Step 1: Understand the Probability of Drawing a Spade

In a standard deck of 52 cards, there are 13 spades. The probability of drawing a spade from the deck is thus \( \frac{13}{52} = \frac{1}{4} \).

Step 2: Determine Probability of Not Drawing a Spade

The probability of not drawing a spade in a single draw is the complement of drawing a spade, which is \( 1 - \frac{1}{4} = \frac{3}{4} \).

Step 3: Calculate Probability of Failing in the First Draw

The probability of not drawing a spade in the first draw is \( \frac{3}{4} \).

Step 4: Calculate Probability of Failing in the Second Draw

Since the card is replaced after each draw, the probability of not drawing a spade in the second draw is also \( \frac{3}{4} \).

Step 5: Calculate Combined Probability of Failing in First Two Draws

For the first two draws to both result in no spades, multiply the probabilities: \( \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} \).

Key Concepts

Drawing Cards

When you're drawing a card from a deck, it's a fascinating exercise in probability. Each time you draw, you randomly pick one card, giving every card an equal chance of being selected. But what happens when you draw multiple times? If you replace the card each time and shuffle the deck, as in our given exercise, the deck becomes fresh again, maintaining the same probabilities for each draw.
In terms of probability, what you draw can significantly influence your strategy and understanding. For instance, the probability of encountering a particular suit or card depends on its presence in the deck. With 13 cards per suit out of 52, the chance of picking, say, a spade, stays consistent when you replace cards and shuffle anew. This ensures the event of drawing remains unaffected by previous draws.

Deck of Cards

Understanding a standard deck of cards is key to solving problems involving probability theory. A standard deck contains 52 cards split evenly into four suits: spades, hearts, diamonds, and clubs. Each suit has 13 cards: Ace through King. Knowing this structure is essential to calculating probabilities.
For example, if we're asked to calculate the probability of drawing a spade, we note there are 13 spades in the deck. Thus, the probability of drawing a spade is:

• \[ \frac{13}{52} = \frac{1}{4} \]

This fundamental breakdown of a deck allows us to understand scenarios of drawing specific cards or outcomes, such as variables like replacing the card. By appreciating the makeup of the deck, you can see how it applies to varied probability challenges, ensuring your calculations remain spot-on.

Independent Events

Independent events are those that have no relationship in terms of results affecting each other. In the context of drawing cards, when you replace a card and shuffle the deck, each draw is independent. This means whatever the result of the first draw, it doesn't affect the second draw's outcome.
For our exercise, not drawing a spade twice involves successive independent events. The event that you don't draw a spade on the first attempt is independent of not drawing it on the second attempt when you replace the card. Hence, to calculate the probability of failing to draw a spade twice, you simply multiply their probabilities:

• Fail first draw: \( \frac{3}{4} \)
• Fail second draw: \( \frac{3}{4} \)
• Combined: \( \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} \)

Highlighting independent events is crucial as it ensures that probability calculations respect the nature of randomness and equal potential for outcomes, allowing accurate assessments in probability theory.