Problem 12 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Cards are drawn from a pack of 52 cards one by one. The probability that exactly 10 cards will be drawn before the first ace is (A) \(\frac{451}{884}\) (B) \(\frac{241}{1456}\) (C) \(\frac{164}{4165}\) (D) none of these

Step-by-Step Solution

Answer

(D) none of these.

Step 1: Understand the Problem

We need to find the probability that exactly 10 cards are drawn before the first ace appears in a standard deck of 52 cards. A standard deck contains 4 aces and 48 non-ace cards.

Step 2: Identify the Probability Formula to Use

This problem can be solved using the negative binomial distribution, where we want the first success (drawing an ace) to occur on the 11th draw. Thus, we consider the probability of 10 "failures" (non-aces) before the first "success" (an ace).

Step 3: Calculate the Probability of Drawing 10 Non-Aces First

The probability of drawing a non-ace card is \( \frac{48}{52} = \frac{12}{13} \). We need this to happen 10 times consecutively.

Step 4: Calculate the Probability of Drawing an Ace as the 11th Card

Once 10 non-aces have been drawn, the probability of drawing an ace as the 11th card is \( \frac{4}{42} = \frac{1}{10.5} \). However, this needs to be carefully considered in context of order completion.

Step 5: Use the Negative Binomial Probability

The general formula for the negative binomial probability (for the r-th success on the n-th trial) is: \[ P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r} \] where r = 1 (first success), k = 11, p (probability of drawing an ace) = \( \frac{4}{52} = \frac{1}{13} \), and (1-p) = \( \frac{12}{13} \).

Step 6: Calculate the Probability using the Formula

Substituting into the formula, we get: \[ P(X = 11) = \binom{10}{0} \left(\frac{1}{13}\right)^1 \left(\frac{12}{13}\right)^{10} \] \[ = 1 \times \left(\frac{1}{13}\right) \times \left(\frac{61917364224}{137858491849}\right) \] which simplifies more but yields a probability consistent with none of the provided answers.

Step 7: Compare with Given Options

The probability calculated is complex and involves understanding correct fraction simplification or validation against given options which points toward (D) none of these given numerical challenges.

Key Concepts

Negative Binomial Distribution

When we deal with events happening in a sequence, the negative binomial distribution can help us. It's very useful in determining when the first 'success' occurs after a certain number of 'failures'. Think of "success" as drawing the card we want, like an ace, and "failure" as drawing other cards, like non-aces.

The negative binomial distribution allows us to calculate the probability of these events occurring in a specific order. In this exercise, we care about drawing one ace after drawing ten other cards first. Here's how it works in steps:

• Set the event for success: drawing an ace
• Count how many failures (non-ace cards) happen before a success: ten in this case
• Determine the probability for each: the probability of drawing a non-ace card, and then an ace

This leads us to use the negative binomial distribution formula to find out how likely this series of events is. This way, even though card games involve luck, mathematics can predict the patterns we expect to see.

Deck of Cards

A standard deck of cards is a familiar framework for many probability problems. It consists of 52 cards divided into four suits: hearts, diamonds, clubs, and spades. In each suit, there are 13 cards, including one ace.

When thinking of probabilities, each card draw represents an independent event until the deck is reshuffled or the game ends. This means each card pulled impacts the probabilities of following draws. A simple fact crucial in card probabilities is:

• 4 aces per deck
• 48 non-aces

Therefore, any problem that uses a deck requires consideration of these basic facts. Understanding how these factors distribute can simplify calculating probabilities for various outcomes.

Combinatorics

Combinatorics is all about counting and arranging. It's a key part of understanding probability, especially in card games. This exercise uses combinatorial thinking to line up our draws correctly. How? By determining how many ways we can draw ten non-aces followed by an ace.

The negative binomial distribution formula incorporates combinatorial elements. We use combinations to figure out different arrangements of successes and failures in our draws.

• Choose which cards count as "failures"
• Calculate arrangements for our sequence (non-aces first, ace second)

In this exercise, understanding combinatorics lets us know how likely particular card sequences are. Through this, the unpredictability of drawing cards becomes more predictable through numbers and arrangements.