Problem 19 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Two dice are thrown together. The probability that sum of the two numbers will be a multiple of 4 is (a) \(1 / 9\) (b) \(1 / 3\) (c) \(1 / 4\) (d) \(5 / 9\)

Step-by-Step Solution

Answer

The probability that the sum is a multiple of 4 is \(\frac{1}{4}\), which is option (c).

Step 1: Determine Possible Outcomes

When two dice are thrown, each die has 6 faces, so there are a total of \(6 \times 6 = 36\) possible outcomes. We need to consider how many of these outcomes result in a sum that is a multiple of 4.

Step 2: Find Sums That Are Multiples of 4

A multiple of 4 from the sum of two dice can be 4, 8, or 12. Let's determine which outcomes yield these sums. For instance, a sum of 4 can occur with the combinations: (1,3), (2,2), (3,1). Check for all sums: - Sum of 4: (1,3), (2,2), (3,1) - Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2) - Sum of 12: (6,6).

Step 3: Count Favorable Outcomes

List out all the favorable outcomes: (1,3), (2,2), (3,1), (2,6), (3,5), (4,4), (5,3), (6,2), (6,6). These are 9 possible outcomes where the sum is a multiple of 4.

Step 4: Calculate Probability

Probability is defined as the ratio of the number of favorable outcomes to the total number of outcomes. So, the probability is \(\frac{9}{36} = \frac{1}{4}\).

Key Concepts

Dice Probability

Rolling dice is a perfect way to understand basic probability concepts. When talking about dice, probability helps us predict the chances of different outcomes when dice are thrown.
Each dice face is equally likely to show up, and since a standard die has 6 faces, each face has a probability of \(: \frac{1}{6} \).
When you throw two dice at once, you increase the number of possible outcomes dramatically, as each die can end up showing any number from 1 to 6.

• This raises the possible outcomes to \(6 \times 6 = 36\)
• All outcomes are equally likely given the randomness of a fair throw.

Dice are often used in probability exercises precisely because the outcomes are easy to list and calculate, making calculations straightforward.

Combinations of Dice Rolls

With two dice, each die acting independently, the possibilities for combinations broaden.
One die can land on any one of its six sides while the second die can also land on any of its six sides.

• This results in 36 possible combinations of rolls, such as (1,1), (1,2), ..., (6,6).
• This comprehensive listing allows us to evaluate how often certain outcomes occur.

A part of understanding combinations is recognizing symmetrical and predictable patterns within them.
For example, the combination (2,3) is considered different from (3,2) even though their sums are identical.
This distinction is crucial in calculating specific outcomes, such as a sum of numbers leading to a specific probability event.

Sum of Two Dice

The sum of two dice is a simple yet fascinating concept in probability. When you roll two dice, their face values are added together to get a sum.
These sums range from 2, the minimum (when both dice show 1), up to 12, the maximum (when both show 6).

• Some sums, like 7, are more common due to more combinations resulting in them, such as (1,6), (2,5), (3,4), etc.
• On the other hand, sums like 12 occur less frequently, as they arise only from (6,6).

The probability of getting a specific sum depends on how many combinations can make that sum.
For example, there are five combinations to achieve a sum of 8, but only one combination for 2 and 12.
Understanding this balance helps predict the outcome frequencies when rolling dice, and these sums are vital to solving dice-related probability problems.

Multiples of Numbers in Probability

One interesting aspect of probability involves determining how often a sum is a multiple of a given number.
In the context of dice, for example, we might be interested in how frequently the sum of two dice is a multiple of 4.

• Multiples of 4 within the expected range (2 to 12) of dice sums are 4, 8, and 12.
• These sums do not occur with equal frequency; for instance, a sum of 4 can appear from combinations like (1,3) or (2,2).

By counting all the outcomes that meet this condition, one can determine the probability of the event.
In our case, nine combinations lead to these sums, meaning the probability is \(\frac{9}{36} = \frac{1}{4}\).
Approaching probability with this method allows for breaking down complex problems into more manageable parts, reinforcing understanding and application.