Problem 112 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A pack of playing cards was found to contain only 51 cards. If the first 13 cards, which are examined, are all red, the probability that the missing card is black is (A) \(\frac{2}{3}\) (B) \(\frac{1}{3}\) (C) \(\frac{2}{9}\) (D) none of these

Step-by-Step Solution

Answer

The missing card is more likely to be black than red under these conditions (1/3).

Step 1: Understand What is Given

We have a pack of 51 cards because one card is missing from the original 52 cards in a normal deck. The first 13 cards that were checked are all red.

Step 2: Identify Possible Outcomes

Since there are 26 red cards in a full deck and all 13 cards checked are red, there are still 13 red cards left in the remaining pile. There are originally 26 black cards, so if the red cards were examined, the remaining unknown 38 cards likely contain all 26 black cards plus 12 red cards.

Step 3: Determine Probable Missing Card Color

Since the pack only has 51 cards, one card is missing. The checker found all 13 cards red, which means if the missing card were red, then there would be 25 out of 51 total cards left. But since 13 of those are black, this means there are 25 red and 13 black in the full set of 51 cards. If we take 1 red out of these available, we'll be left with 12, leading to possibility of red missing = \(1-\frac{25}{3+25}\).

Key Concepts

Playing Cards

Playing cards are a common element in probability exercises. A standard deck contains 52 cards, divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, ranging from Ace to King. In the context of this problem, the red cards include hearts and diamonds, while black cards include clubs and spades. This structure forms the basis for many probability questions, as it offers a balanced and controlled environment. Understanding the makeup of a deck is crucial when solving card-related probability problems. You need to consider the total number of cards, how they are distributed across suits and colors, and what each subset of cards represents in terms of probability outcomes. This thorough comprehension permits you to effectively analyze the situations like the missing card problem we're discussing.

Missing Card Problem

The missing card problem is a unique type of probability question that adds complexity by altering the usual set of circumstances. Here, instead of dealing with a standard 52-card deck, you have only 51 cards because one card is missing. This change requires careful adjustment of probabilities because we no longer have a complete representation of the deck. To tackle this kind of problem, you should first establish what remains known, such as the total number of red and black cards initially. Then, deduce how the missing card affects these quantities. In our example, discovering that the first 13 are red helps narrow down the possibilities for what the missing card might be.

Conditional Probability

Conditional probability is key to solving problems where there's a dependency between events. In this context, the probability of the missing card being black depends on the fact that all 13 inspected cards were red.Conditional probability allows you to update your initial estimates upon obtaining new information. mathematically, it's represented as:\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]where \(P(A|B)\) is the probability of event A occurring, given that B has occurred. Applying this to the problem, once you verify that 13 cards are red, your focus shifts to how the 38 remaining cards (exempting the missing one) affect the outcome. This helps you better calculate the likelihood of the missing card's color and leads you to the right solution.

Red and Black Cards

Red and black cards are specifically discussed to highlight their relevance in this probability problem. In a normal deck, there are 26 red cards and 26 black cards, making an equal split. But, with a missing card, this balance might be disturbed. Knowing that the first 13 cards were all red reduces the number of red cards in the rest of the deck, impacting the probability calculations. It suggests that the pile of unseen cards now mostly contains black cards. As part of your solution, assessing the proportions of red and black remaining becomes vital to determine the color of the missing card. Through keen observation and understanding of these groupings, you can deduce the probability that the absent card is black, which is a critical exercise in prediction under varying circumstances.