JEE Advanced 1994

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Animated Solution for Mathematics - Probability: 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 shuffled pack of eleven cards numbered 2, 3, 4, ..., 12 is picked and the number on the card is noted. What is the probability that the noted number is either 7 or 8?

Enter Numerical Value:

Visualized Solution

The Initial Setup: Coin Toss

Initial event: Tossing an unbiased coin.
Outcomes: Head ( $H$ ) or Tail ( $T$ ).
Probabilities: $P (H) = \frac{1}{2}$ and $P (T) = \frac{1}{2}$ .

Path 1: Getting a Head

If the result is a Head ( $H$ ), we roll two unbiased dice.
Total outcomes for two dice = $6 \times 6 = 36$ .

Favorable Outcomes for Dice

Target: Sum of numbers on the dice is $7$ or $8$ .
Pairs for sum $7$ : $(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)$ $\to 6$ outcomes.
Pairs for sum $8$ : $(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)$ $\to 5$ outcomes.

Probability for Dice Path

Total favorable outcomes = $6 + 5 = 11$ .
Conditional Probability: $P (Sum 7 or 8 ∣ H) = \frac{11}{36}$ .

Path 2: Getting a Tail

If the result is a Tail ( $T$ ), we pick a card from a well-shuffled pack.
The pack contains $11$ cards numbered $2, 3, 4, \dots, 12$ .

Favorable Outcomes for Cards

Target: Number on the picked card is $7$ or $8$ .
Favorable cards: ${7, 8}$ $\to 2$ outcomes.

Probability for Card Path

Conditional Probability: $P (Card 7 or 8 ∣ T) = \frac{2}{11}$ .

The Law of Total Probability

Using the Law of Total Probability to combine both cases:
$P (Total) = P (H) \cdot P (Sum ∣ H) + P (T) \cdot P (Card ∣ T)$

Substituting the Values

Substitute the calculated probabilities into the formula:
$P = \frac{1}{2} (\frac{11}{36}) + \frac{1}{2} (\frac{2}{11})$

Algebraic Simplification

Simplify the individual terms:
$P = \frac{11}{72} + \frac{1}{11}$

Final Calculation

Find the common denominator and add:
$P = \frac{11 \times 11 + 72 \times 1}{72 \times 11} = \frac{121 + 72}{792}$
$P = \frac{193}{792} \approx 0.2436$

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The Sigma Insight: Total Probability Theorem

Solution Diagram

Analyzing the Setup

Imagine you are standing at a crossroads, holding a perfectly fair coin. When you toss this unbiased coin, the universe splits into two equally likely paths.

Since the coin is fair, the probability of getting a Head is exactly

P (H) = \frac{1}{2}

, and the probability of getting a Tail is

P (T) = \frac{1}{2}

Path 1

The Dice Universe

If the coin lands on Heads, we roll a pair of unbiased dice. The total number of outcomes in this sample space is

6 \times 6 = 36

We seek a sum of either seven or eight. For a sum of seven, the possible pairs are

(1, 6), (2, 5), (3, 4), (4, 3), (5, 2),

and

(6, 1)

, totaling 6 outcomes. For a sum of eight, the pairs are

(2, 6), (3, 5), (4, 4), (5, 3),

and

(6, 2)

, totaling 5 outcomes.

The total number of favorable outcomes is

6 + 5 = 11

. Thus, the conditional probability is:

P (Sum 7 or 8 ∣ H) = \frac{11}{36}

Path 2

The Card Universe

If the coin lands on Tails, we pick a single card from a pack numbered 2 through 12. The total number of cards is 11.

Our winning condition is picking a card numbered 7 or 8. There are exactly 2 such cards in the pack.

The conditional probability for this branch is:

P (Card 7 or 8 ∣ T) = \frac{2}{11}

The Synthesis

Law of Total Probability

To find the overall probability, we use the Law of Total Probability. This theorem combines the mutually exclusive events into the following equation:

P (Total) = P (H) \cdot P (Sum ∣ H) + P (T) \cdot P (Card ∣ T)

Substituting our calculated values, we get:

P = \frac{1}{2} (\frac{11}{36}) + \frac{1}{2} (\frac{2}{11})

The Final Calculation

Simplifying the terms, the first part becomes

\frac{11}{72}

and the second part simplifies to

\frac{1}{11}

. We now add these fractions:

P = \frac{11}{72} + \frac{1}{11}

To add these, we find the common denominator, which is

72 \times 11 = 792

. Performing the cross-multiplication:

P = \frac{11 \times 11 + 72 \times 1}{792} = \frac{121 + 72}{792}

The final probability is:

$P = \frac{193}{792} \approx 0.2436$

Common Traps & FAQs

Wait, why do we add the probabilities instead of multiplying them at the end?

That is a fantastic question! We add them because the two scenarios (getting a Head and getting a Tail) are mutually exclusive. You cannot flip both a Head and a Tail in a single toss. The Law of Total Probability states that if you have a set of mutually exclusive events that cover all possibilities, you sum the probabilities of the event occurring within each of those scenarios, weighted by the probability of that scenario happening itself.

Why is the sample space for the dice 36, but for the cards it is 11?

The sample space depends entirely on the experiment being performed. When you roll two dice, each die has 6 faces, so the total combinations are

6 \times 6 = 36

. However, the card experiment is different; you are picking one card from a specific set of 11 cards numbered 2 through 12. The sample space is simply the total number of cards available, which is 11. Always define your sample space based on the specific action being taken!

Is there any overlap between the two events? Can I get a Head and a Tail?

No, the coin toss is a binary event. It is either a Head or a Tail. Because these outcomes are mutually exclusive and collectively exhaustive, we treat the two branches as completely separate paths. There is no overlap, which is exactly why the Law of Total Probability works so elegantly here.