Problem 28 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

One card is drawn from the pack of cards. It is now replace in the pack and a card is again drawn. If it is done six times, then the probability of coming 2 cards of heart, 2 of diamond and 2 red cards in order is (a) \((1 / 4)^{4}\) (b) \((1 / 4)^{5}\) (c) \((1 / 4)^{6}\) (d) None of these

Step-by-Step Solution

Answer

(a) \((1 / 4)^{4}\)

Step 1: Determine the number of each card type in the deck

A standard deck consists of 52 cards, with 13 hearts and 13 diamonds, and 26 cards are red (hearts and diamonds combined).

Step 2: Define the scenario for drawing cards

We need to find the probability of drawing exactly 2 hearts, 2 diamonds, and 2 red cards in 6 draws, in that specific order.

Step 3: Calculate the probability for each type

The probability of drawing a heart is \( \frac{13}{52} = \frac{1}{4} \), a diamond is \( \frac{13}{52} = \frac{1}{4} \), and a red card (which is either a heart or diamond) is also \( \frac{26}{52} = \frac{1}{2} \).

Step 4: Compute the total probability for the sequence

Multiply the probabilities for the sequence of first two hearts, then two diamonds, then two red cards: \( \left(\frac{1}{4}\right)^2 \times \left(\frac{1}{4}\right)^2 \times \left(\frac{1}{2}\right)^2 \).

Step 5: Simplify the expression

The expression becomes \( \frac{1}{16} \times \frac{1}{16} \times \frac{1}{4} = \frac{1}{1024} \).

Step 6: Check the options and conclude

The probability calculated, \( \frac{1}{1024} \), is equivalent to \( \left(\frac{1}{4}\right)^4 \). Thus, the correct option is (a) \( \left(\frac{1}{4}\right)^4 \).

Key Concepts

Probability Calculations

Probability calculations help us understand how likely certain events are to happen. In card games, this can involve figuring out the chance of drawing specific cards. When you draw a card from a deck, put it back, and draw again, it's called a draw with replacement. This keeps the probabilities the same each time because the deck stays full and unchanged. To find the probability of drawing a specific sequence of cards, multiply the individual probabilities in sequence. For example:

• Probability of drawing a heart: \( \frac{13}{52} = \frac{1}{4} \)
• Probability of drawing a diamond, similarly: \( \frac{1}{4} \)
• For a red card (either heart or diamond): \( \frac{26}{52} = \frac{1}{2} \)

To find the probability of a whole sequence, you multiply the probabilities of each event happening in a row.

Combinatorics

Combinatorics is a fantastic tool to solve probability problems by counting the different ways we can achieve a particular outcome. When dealing with cards, combinatorics might help us determine how many ways we can draw a particular sequence of cards. In our example, we are interested in a particular order: two hearts, two diamonds, and two red cards. Combinatorics doesn't just give us the number of ways; it helps us understand the structure of this sequence:

• For two specific cards (like two hearts), only that subset counts.
• For just red cards, it could be any combination of hearts and diamonds adding up to the total desired number.

So when you combine combinatorics with probability, you essentially calculate the chances over all the structurally similar possible outcomes.

Deck of Cards

A deck of cards is a common tool in many probability exercises. Understanding its structure is crucial for solving related problems effectively. A standard deck contains 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, which is fundamental for calculating probabilities. For color-specific inquiries, remember:

• Hearts and diamonds are red (26 cards total).
• Clubs and spades are black (also 26 cards total).

Knowing this structure helps us calculate probabilities by clearly identifying the number of favorable outcomes out of the total possible outcomes. When tasked with a sequence or a combination of draws, recognizing the deck's setup enables targeted calculations, like drawing specific numbers of each suit or color.