Problem 19 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shufled pack of eleven cards numbered \(2,3,4, \ldots, 12\) is picked and thenumber on the card is noted. The probability that the noted number is either 7 or 8 , is (A) \(\frac{193}{792}\) (B) \(\frac{164}{792}\) (C) \(\frac{231}{792}\) (D) none of these

Step-by-Step Solution

Answer

The answer is not listed in the options.

Step 1: Determine Possible Outcomes for Coin Toss

The coin is unbiased, so there are two possible outcomes: heads (H) or tails (T). The probability for each is \( \frac{1}{2} \).

Step 2: Probabilities for Rolling Dice After Getting a Head

If a head occurs, we roll two dice. Each die has 6 faces, so there are a total of 36 possible outcomes when rolling two dice. We add the numbers on the two dice for each outcome.

Step 3: Identify Favorable Outcomes for Dice Roll

The sums that can be 7 or 8 are as follows: \(7 = (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\) - 6 ways; \(8 = (2,6), (3,5), (4,4), (5,3), (6,2)\) - 5 ways. So, there are 11 favorable outcomes (6 for 7 and 5 for 8).

Step 4: Probability of Getting 7 or 8 When Dice Rolled

The probability of rolling a sum of 7 or 8 with two dice is \( \frac{11}{36} \).

Step 5: Probability of Picking a Certain Card After Getting a Tail

If a tail occurs, a card is picked from 11 cards numbered \(2, 3, 4, \ldots, 12\). There are two favorable numbers for 7 and 8, specifically the cards 7 and 8. The probability of picking these is \( \frac{2}{11} \).

Step 6: Calculate the Total Probability of Getting a 7 or 8

The total probability is the sum of probabilities of heads times the probability of dice showing 7 or 8, and tails times the probability of picking card 7 or 8. This becomes:\[ \frac{1}{2} \times \frac{11}{36} + \frac{1}{2} \times \frac{2}{11} \].

Step 7: Simplify the Expression and Find Solution

Calculate the overall probability:1. \( \frac{1}{2} \times \frac{11}{36} = \frac{11}{72} \)2. \( \frac{1}{2} \times \frac{2}{11} = \frac{1}{11} = \frac{6}{66} = \frac{6}{72} \)3. Add: \( \frac{11}{72} + \frac{6}{72} = \frac{17}{72} \)Convert to a common denominator of 792: Multiply by 11:\( \frac{17}{72} \times \frac{11}{11} = \frac{187}{792} \) which does not match any of the provided options.

Key Concepts

Dice Rolling

Rolling dice is a classic exercise in probability, due to the predictability of outcomes and simplicity in calculation. When you roll a single six-sided die, there are six outcomes, each with a probability of \(\frac{1}{6}\). However, rolling two dice increases the number of potential outcomes exponentially to 36, because each die operates independently with 6 outcomes. Whenever the sum of the numbers on the two dice is important (which is a common scenario), understanding how these numbers arise is crucial:

• Sums range from 2 (1+1) to 12 (6+6).
• Different sums are achieved in different ways: for example, a sum of 7 can be rolled as (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1) — 6 different combinations.

Knowing these combinations helps determine probabilities. For example, the probability of rolling a sum of 7 or 8 on two six-sided dice is checked by listing all pairs that contribute to these sums and counting them. This probability is then the ratio of favorable outcomes over total possibilities: \(\frac{11}{36}\).

Card Deck Probabilities

Card probabilities often mirror concepts learned with dice and provide a rich field for refined probability exercises. In our example, a unique deck of cards numbered from 2 to 12 introduces a straightforward setup: drawing numbers with equal likelihood. Imagine these cards laid out in front of you. Each number from 2 to 12 has one card each, making a total of 11 cards. If you wish to find the probability of drawing a card with a number, say 7 or 8, two favorable outcomes exist. Therefore, the probability becomes ** \(\frac{2}{11}\).** This scenario exemplifies basic probability: comparing favorable outcomes to the total. Unlike rolling dice, here outcomes do not compound. This means each card is a singular event. When combined with other probability events, such as a coin toss or dice roll, these singular events can build into more complex probability problems.

Unbiased Coin

The concept of an unbiased coin toss illustrates the simplest form of probability. When a coin is tossed, there are two possible outcomes: heads or tails, each happening with a probability of \(\frac{1}{2}\) since the coin is fair and unbiased.This scenario reflects an important principle in probability: **independence.** The result of one toss does not affect the next; each flip of the coin resets the probability. In combined probability settings, understanding the independence of a coin toss helps calculate total probabilities. For instance, if a problem includes an unbiased coin directing subsequent actions (e.g., rolling dice or drawing a card), each branch of possible events must be calculated separately:

• Coin outcome `heads` could lead to rolling dice and calculating those probabilities.
• Coin outcome `tails` may lead to a card draw and those associated probabilities.

Finally, all these paths are combined based on the initial coin probability, keeping calculations straightforward and accurate.