Problem 19 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered \(1,2,3, \ldots, 9\) is randomly picked and the number is èitheer 7 or 8 is: (a) \(\frac{13}{36}\) (b) \(\frac{15}{72}\) (c) \(\frac{19}{72}\) (d) \(\frac{19}{36}\)

Step-by-Step Solution

Answer

The correct answer is (c) \(\frac{19}{72}\).

Step 1: Understand the Probability of Coin Toss

The coin is unbiased, meaning there is an equal chance of getting a head or a tail. Thus, the probability of getting either a head or a tail is \( \frac{1}{2} \).

Step 2: Calculate Probability for Sum (7 or 8) with Dice

When a head is tossed, two unbiased dice are rolled. To find the probability of the sum being 7 or 8, compute:- The sum 7 can occur in 6 ways: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).- The sum 8 can occur in 5 ways: (2,6), (3,5), (4,4), (5,3), and (6,2).So, there are a total of 11 favorable outcomes for either 7 or 8 out of 36 possible outcomes (as each die has 6 faces, giving \(6 \times 6 = 36\) combinations).Thus, the probability of rolling a 7 or 8 is \(\frac{11}{36}\).

Step 3: Calculate Probability to Pick Card 7 or 8

If a tail is the result of the toss, a card is drawn from a set numbered 1 to 9. The cards of interest are 7 and 8, which means there are 2 favorable outcomes. Therefore, the probability of picking a 7 or 8 is \(\frac{2}{9}\).

Step 4: Compute Total Probability for 7 or 8

We combine these probabilities based on the toss result:- Probability of obtaining a sum of 7 or 8 after tossing a head is \(\frac{1}{2} \times \frac{11}{36} = \frac{11}{72}\).- Probability of drawing card 7 or 8 after tail is \(\frac{1}{2} \times \frac{2}{9} = \frac{1}{9} = \frac{8}{72}\).Add both probabilities: \(\frac{11}{72} + \frac{8}{72} = \frac{19}{72}\).

Step 5: Solution Verification

Reviewing the calculations and problem ensures that each step logically follows the previous one. Both parts of the problem (dice with heads and cards with tails) separately contribute to the probability of 7 or 8, correctly summed to ensure the correct process.

Key Concepts

Unbiased Coin

An unbiased coin is a coin that has no preference for landing on heads or tails. In probability terms, this means that the chance of the coin landing on heads is exactly the same as the chance of it landing on tails. When we toss an unbiased coin, we expect both outcomes to occur equally over a large number of tosses.
The probability of getting either a head or a tail in a single toss is therefore \(\frac{1}{2}\). This forms the basis for many probability problems, as an unbiased coin provides a straightforward 50-50 chance of each outcome.

Dice Probability

Dice probability refers to the likelihood of different outcomes when rolling dice. Each die has six faces, numbered 1 through 6, meaning each face has an equal probability of landing face up.
This probability is \(\frac{1}{6}\) for any single outcome (e.g., rolling a 3). When rolling two dice, there are \(6 \times 6 = 36\) possible outcomes.

• For example, to get a sum of 7, you have six combinations: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
• To get a sum of 8, there are five combinations: (2,6), (3,5), (4,4), (5,3), and (6,2).

Thus, the probability of rolling a sum of 7 or 8 with two dice is calculated by adding the probabilities of each sum. This gives: \( \frac{11}{36} \), as there are 11 favorable combinations out of 36 possible ones.

Card Probability

In situations where cards are involved, such as picking a card from a deck, the probability depends on the total number of cards and the number of favorable outcomes.
For a set of nine cards numbered 1 through 9, each card has an equal probability of being picked, that is \( \frac{1}{9} \).
If the task is to draw either a 7 or an 8, there are two favorable choices (7 and 8), making the probability of drawing one of these specific cards \( \frac{2}{9} \). This straightforward calculation is a basic example of calculating probability with cards.

Sum of Dice

The sum of two dice is a classic probability problem. With each die having six faces numbered from 1 to 6, the smallest sum is 2 (1+1) and the largest is 12 (6+6).
The probability of achieving any specific sum depends on how many combinations can yield that sum.
For a sum of 7 or 8, which are among the more frequently occurring sums, identifying the number of combinations helps calculate their probabilities:

• Sum of 7: 6 combinations.
• Sum of 8: 5 combinations.

Each sum's probability is determined by dividing the number of favorable outcomes by the total possible outcomes, 36 (since each die shows one face, \(6 \times 6 = 36\)).

Random Selection

Random selection is a process where each element of a set has an equal chance of being chosen. This forms the cornerstone of statistical experiments and objective sampling methods.
For instance, drawing a card from a shuffled deck or tossing a coin introduces randomness, ensuring that no outcome is overly dependent on previous events. It’s important for ensuring fairness and balance in probability exercises.
When dealing with the cards numbered 1 to 9, random selection guarantees that picking occurs without bias, making calculating probability a straightforward exercise.

Favorability in Probability

Favorability in probability refers to the way outcomes are weighed when calculating probabilities. It concerns the number of successful or 'favorable' outcomes within the total set of possibilities.
To determine favorability, count how many ways a particular event can occur - these are your 'favorable' outcomes.
For example, discovering the probability of tossing a coin for a head involves one favorable outcome (the head) out of two possible outcomes, resulting in a probability of \( \frac{1}{2} \).

• For dice, finding the sum of 7 or 8 involves counting all combinations that can produce these sums as mentioned previously.
• In card selection, determining how likely it is to pick a 7 or 8 involves two favorable cards out of the nine.

This straightforward understanding helps clarify the likelihood of events, simplifying the art of probability.