Problem 18 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A pair of unbiased dice is rolled together till a sum of either 5 or 7 is obtained. The probability that 5 comes before 7 is (a) \(1 / 5\) (b) \(2 / 5\) (c) \(3 / 5\) (d) \(4 / 5\) (b) Let A: 'event that sum is 5 ' \(B:\) 'event that sum is 7' \(C:\) 'event that sum is neither 5 nor 7 ' Then \(P(A)=\frac{4}{36}=\frac{1}{9}, P(B)=\frac{6}{36}=\frac{1}{6}\) \(P(C)=\frac{26}{36}=\frac{13}{18}\) Now probability that \(A\) occurs before \(B\) $$ \begin{aligned} &=P(A+C A+C C A+\ldots) \\ &=P(A)+P(C A)+P(C C A)+\ldots \\ &=P(A)+P(C) P(A)+P(C) P(C) P(A)+\ldots \\ &=\frac{1}{9}+\left(\frac{13}{18}\right) \frac{1}{9}+\left(\frac{13}{18}\right)^{2} \frac{1}{9}+\ldots \\ &=\frac{1 / 9}{1}=\frac{2}{5} \end{aligned} $$

Step-by-Step Solution

Answer

The probability is \( \frac{2}{5} \).

Step 1: Understanding the Events

We are rolling a pair of unbiased dice until we get either a sum of 5 or a sum of 7. Let event A be the sum being 5, event B be the sum being 7, and event C be any other sum. We are interested in calculating the probability that the dice result in a sum of 5 before achieving a sum of 7.

Step 2: Calculating Probabilities for A, B, and C

The probability of rolling a sum of 5 (event A) is given by the ratio of the favorable outcomes to total outcomes, which is \( \frac{4}{36} \) since there are 4 combinations to make a sum of 5. Similarly, the probability of getting a sum of 7 (event B) is \( \frac{6}{36} \), and the probability of any other sum (event C) is \( \frac{26}{36} \). Simplifying gives \( P(A) = \frac{1}{9} \), \( P(B) = \frac{1}{6} \), and \( P(C) = \frac{13}{18} \).

Step 3: Expressing the Probability Sequence

The probability that event A occurs before event B can be expressed as the sum of infinite subsequences: A, CA, CCA, etc. This sequence represents A happening immediately, A happening after one C, after two Cs, and so on.

Step 4: Calculating the Infinite Sum

The sequence is an infinite geometric series where the first term is \( P(A) = \frac{1}{9} \) and the common ratio is \( P(C) \). So the probability that A occurs before B, given by this series, is the sum \( \frac{1}{9} + \left(\frac{13}{18}\right) \frac{1}{9} + \left(\frac{13}{18}\right)^2 \frac{1}{9} + \ldots \).

Step 5: Solving the Geometric Series

The sum of an infinite geometric series \( a, ar, ar^2, \ldots \) is given by \( \frac{a}{1-r} \) where \( |r| < 1 \). Here, \( a = \frac{1}{9} \) and \( r = \frac{13}{18} \). Thus, the sum is \( \frac{1/9}{1 - 13/18} = \frac{1/9}{5/18} = \frac{1/9 \times 18/5} = \frac{2}{5} \).

Step 6: Conclusion

The probability that a sum of 5 comes before a sum of 7 when rolling two dice is \( \frac{2}{5} \).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Geometric series are significant in probability, especially when calculating probabilities of sequences of events that repeat under certain conditions.
In the context of the dice probability problem, the series starts with the probability of one event happening (rolling a sum of 5), followed by the same event happening again after one or more other events (rolling sums other than 5 or 7 before a 5). This forms an infinite geometric series where each term is the product of the probability of the preceding events and the probability of the next occurrence of the main event.
To find the probability that the sum of the dice is 5 before 7, we express the repetitive nature of the event sequence as a geometric series.

Dice Probability

Dice probability refers to the likelihood of a specific outcome occurring when rolling dice. Since a die has 6 sides, each with an equal chance of landing face up, the basic probability for any particular face to show is 1/6.
When rolling two dice, various sums can occur, ranging from 2 to 12. The sum of 5, for instance, can be achieved in 4 different ways (e.g., 1+4, 2+3, 3+2, 4+1), making the probability of rolling a sum of 5 equal to \( \frac{4}{36} \). A sum of 7 can be achieved in 6 different combinations, leading to a probability of \( \frac{6}{36} \) or \( \frac{1}{6} \).
Understanding these probabilities is crucial when calculating the likelihood of one specific sum occurring before another, such as determining the sequence in which specific sums are rolled when a pair of dice is repeatedly thrown.

Probability of Events

Probability of events is a fundamental concept in probability theory, involving the likelihood of various possible outcomes occurring. In our scenario, this revolves around distinct outcomes when rolling two dice.
For events defined in the exercise, each event A, B, or C has a specific probability based on dice outcomes. The probability of event A (sum of 5) is \( \frac{1}{9} \), event B (sum of 7) is \( \frac{1}{6} \), and event C (neither sum of 5 nor 7) is \( \frac{13}{18} \).
Calculating composite probabilities involves understanding sequences of these events, such as A occurring first or after several Cs. This sequence relies on combining these events' probabilities to compute probabilities for complex scenarios, like deciding which specified outcome will occur first.

Infinite Series Summation

In mathematics, an infinite series summation is the process of adding an infinite sequence of numbers. This concept is vital for solving problems where events repeat indefinitely, such as dice rolls in probability problems.
The solution to our exercise leverages an infinite geometric series. Here, the sum of a series of probabilities is calculated, where we assume numerous occurrences of sum C happening before achieving sum A or B. Each term in the series is composed of the probability of the ongoing sequence of events, multiplied by the likelihood of the main event happening afterward.
The formula for an infinite geometric series, \( a + ar + ar^2 + \ldots \), is known to be \( \frac{a}{1-r} \) when the common ratio \( r \) is less than 1. Applying this formula enables us to find the probability of achieving a sum of 5 before a sum of 7, as it sums up all possible sequences in a straightforward mathematical expression.