Problem 2 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A player draws a playing card from a set of playing cards. What will be the probability of not being a diamond card? \([\mathrm{MP}-2004(\mathrm{~A})]\)

Step-by-Step Solution

Answer

The probability of not drawing a diamond card is \( \frac{3}{4} \).

Step 1: Understand the Deck

A standard deck of playing cards consists of 52 cards which include four suits: spades, clubs, hearts, and diamonds. Each suit has 13 cards.

Step 2: Total Diamonds

Count the total number of diamond cards in the deck. Since there are four suits and 52 cards total, each suit (including diamonds) has 13 cards.

Step 3: Calculate Non-Diamond Cards

Subtract the number of diamond cards from the total number of cards. That is 52 - 13 = 39 non-diamond cards.

Step 4: Probability of Not Drawing a Diamond

The probability of not drawing a diamond card is the ratio of non-diamond cards to the total number of cards. Use the formula \( P( ext{not a diamond}) = \frac{39}{52} \). Simplifying this fraction gives \( \frac{3}{4} \).

Key Concepts

Playing Cards

A standard deck of playing cards is a popular and widely recognized tool used for card games and mathematical exercises. Typically, a deck consists of 52 cards. These cards are divided into four distinct suits: spades, clubs, hearts, and diamonds.

• Each suit contains exactly 13 cards, which include the numbers 2 through 10, an ace, and the face cards—jack, queen, and king.
• The hearts and diamonds suits are colored red, while the spades and clubs suits are black.

Knowing the structure of a deck is essential, especially when tackling problems in probability theory as it allows for a clear understanding of how many outcomes or scenarios are possible. Understanding these basics ensures that you're starting on the right foot when engaging with probability problems.

Probability Calculation

Probability is an intriguing branch of mathematics that measures the likelihood of an event occurring. When you're working with playing cards, probability calculations help determine the chances of drawing specific cards or avoiding certain ones.
To find the probability of an event:

• Determine the total number of possible outcomes.
• Find the number of favorable outcomes for the specific event.
• Use the formula: Probability = \(\frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\)

In the context of the exercise, the probability of drawing a non-diamond card involves calculating the ratio of non-diamond cards (39) to the total number of cards (52). Simplifying the fraction \(\frac{39}{52}\) gives \(\frac{3}{4}\), meaning there's a 75% chance of drawing a card that is not a diamond.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It's especially useful in probability theory when trying to determine how many ways a particular outcome can occur.
In the scenario of playing cards, combinatorics helps us:

• Consider the arrangement and selection of cards without replacement.
• Evaluate how many possible ways there are to select a subset of cards, such as finding the total number of non-diamond cards.

By using combinatorial strategies, one can simplify complex probability calculations, making it easier to arrive at precise answers. This approach provides a structured way of tackling not only simple problems like calculating the probability of drawing a specific card but also more intricate questions involving multiple steps or conditions.