Question

Two dice are rolled one after the other. The probability that the number on the first is smaller than the number on the second is (a) \((1 / 2)\) (b) \((7 / 18)\) (c) \((3 / 4)\) (d) \((5 / 12)\)

Short answer

The probability that the number on the first dice is smaller than the number on the second dice is \(\frac{5}{12}\).

Step-by-step solution

Determine the sample space of rolling two dice

The sample space (S) of rolling two dice can be represented as pairs of numbers (a, b) where a and b are the numbers on the first and the second dice respectively. Since each dice has 6 faces, there are a total of 6×6 = 36 possible outcomes.

Determine the favorable outcomes

To find the favorable outcomes, we need to identify all the pairs (a,b) where the number on the first dice (a) is smaller than the number on the second dice (b). List down all the favorable outcomes: (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 3), (2, 4), (2, 5), (2, 6) (3, 4), (3, 5), (3, 6) (4, 5), (4, 6) (5, 6) There are 15 favorable outcomes in total.

Calculate the probability

Now we can calculate the probability that the number on the first dice is smaller than the number on the second dice. The probability (P) can be calculated using the formula: P = (Number of favorable outcomes) / (Total outcomes in the sample space) P = 15/36 P can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3: P = (15/3) / (36/3) P = 5/12 The probability that the number on the first dice is smaller than the number on the second dice is \(\frac{5}{12}\). So the correct answer is (d) \(\frac{5}{12}\).

Key concepts

Sample Space

When we roll two dice, delving into the concept of a sample space is fundamental to understand the basics of probability. At its simplest, the sample space is the set of all possible outcomes. In our case, the sample space consists of all the possible pairs of numbers you could get by rolling two dice. Since each die has six sides, and each side is equally likely to land face up, we have a total of six options for the first die and six for the second die, giving us a sample space with 6 x 6, which equals 36 possible outcomes.

Visualizing the sample space can be done by considering it as a grid, where one axis represents the outcomes of the first die and the other axis for the second die. Each cell in the grid represents a unique pair of outcomes. By fully understanding the sample space, students can have a much easier time identifying which outcomes count towards the probability they are trying to calculate.

Favorable Outcomes

The next step in our journey through basic probability concepts brings us to the idea of favorable outcomes. These are the specific outcomes from the sample space that meet the criteria we are interested in. For our exercise, we're focusing on instances where the number on the first die is smaller than the number on the second die.

We can approach this systematically by considering the first die and listing the second die's possible numbers that would make the outcome favorable. For instance, if the first die shows a 1, any number from 2 to 6 on the second die would fulfill our condition, which gives us five favorable outcomes. Repeating this process for each number on the first die provides us a clear picture of all the possible winning combinations. There's a pattern we can observe: as the number on the first die increases, the number of favorable outcomes decreases, leading us to a total of 15 favorable outcomes.

Probability Calculation

Now, let's combine our understanding of sample space and favorable outcomes to calculate probability. The probability calculation is a mathematical way of expressing the likelihood of an event occurring. The formula we use is simple:\[\begin{equation}P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes in the sample space}}\end{equation}\]

In our dice problem, we have 15 favorable outcomes and 36 possible outcomes in total. Plugging these numbers into our formula gives us a probability of \[\begin{equation}P = \frac{15}{36}\end{equation}\] which simplifies to \[\begin{equation}P = \frac{5}{12}\end{equation}\]. This calculation tells us that if you roll two dice, the probability of the first showing a smaller number than the second is \[\begin{equation}\frac{5}{12}\end{equation}\]. Remember, understanding each part of the equation is crucial for grasping the concept of probability!