Question

A dice is loaded so that the probability of face \(\mathrm{i}\) is proportional to i. \(\mathrm{i}=1,2, \ldots .6\). Then the probability of an even number occupy when the dice is rolled is (a) \((2 / 7)\) (b) \((3 / 7)\) (c) \((4 / 7)\) (d) \((5 / 7)\)

Short answer

The probability of getting an even number when rolling the loaded dice is \(\frac{4}{7}\).

Step-by-step solution

Determine the Constant of Proportionality

The problem states that the probability of each face i is proportional to i. We can write this relationship as: \(P(i) = ki\) where P(i) is the probability of face i, k is the constant of proportionality, and i is the face number. We know that the sum of all probabilities for faces 1, 2, 3, 4, 5, and 6 must equal 1 because these represent all possible outcomes: \(P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1\) \(+ k + 2k + 3k + 4k + 5k + 6k = 1\) \(+ 21k = 1\) Solving for k, we find that: \(k = \frac{1}{21}\) Now we have the constant of proportionality, which will help us find the probability for each even face.

Calculate Probability for Even Faces

Using the constant of proportionality, we can find the probability for each even face (2, 4, and 6): \(P(2) = k \cdot 2 = \frac{1}{21} \cdot 2 = \frac{2}{21}\) \(P(4) = k \cdot 4 = \frac{1}{21} \cdot 4 = \frac{4}{21}\) \(P(6) = k \cdot 6 = \frac{1}{21} \cdot 6 = \frac{6}{21}\)

Find the Total Probability of Getting an Even Number

Now we add the probabilities for faces 2, 4, and 6 to find the total probability of getting an even number: \(P(\text{even}) = P(2) + P(4) + P(6) = \frac{2}{21} + \frac{4}{21} + \frac{6}{21}\) \(P(\text{even}) = \frac{2+4+6}{21} = \frac{12}{21}\) Simplifying the fraction, we find that: \(P(\text{even}) = \frac{12}{21} = \frac{4}{7}\) So, the probability of getting an even number when rolling the dice is \(\frac{4}{7}\), which corresponds to answer choice (c).

Key concepts

Constant of Proportionality

When we're dealing with a loaded dice – one that doesn't give each number the same chance of landing face up – understanding the 'constant of proportionality' becomes essential. In probability, proportionality indicates that there is a direct relationship between two quantities. For our dice, it means the likelihood of a number appearing is directly tied to the number itself.

In our case, the probability of each die face, denoted by P(i), increases with the face number, i. Mathematically, this is expressed as P(i) = ki, where k is the constant of proportionality. To find k, we look at the inherent rule of probabilities – that they must all sum to 1, since one side must land face up every single roll. That's our mathematical certainty.

In the dice example, we add up all the individual face probabilities (which are multiples of k) and set the sum equal to 1. This allows us to solve for k, providing us the foundation to calculate the probability of each specific event, like landing an even number. This core concept is pivotal for it enables us to transition from proportionality to actual probabilities in a loaded dice scenario.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena and modelling the likelihood of different outcomes. At its most fundamental level, it's about making educated guesses regarding the outcomes of various events, like rolling our loaded dice.

The central principle in probability is that the probabilities of all possible outcomes must add up to 1. This is because one of the outcomes must occur, making them collectively exhaustive. For the loaded dice, probability theory guides us in understanding not only the likelihood of each individual number but the composite probability of events, such as rolling an even or odd number. With this insight, we can make predictions and even strategize if we were to use this dice in a game. Knowing the basic constructs of probability theory allows us to approach scenarios involving chance with a quantitative perspective.

Sum of Probabilities

A foundational concept in understanding any probabilistic scenario is recognizing the 'sum of probabilities'. This concept states that when you're considering all possible exclusive outcomes of a probabilistic event, their probabilities must add up to 1. In simpler terms, if you're tossing a coin, the chances of landing heads or tails – the only two outcomes – must equal 100%.

For our loaded dice, each face 1 through 6 represents an exclusive and exhaustive outcome. Therefore, by adding up the individual probabilities of each face and setting their sum equal to 1, we find the constant that helps us make sense of the entire probability model. This sum allows us to verify our work and encapsulates the notion that all potential results have been accounted for. The sum of probabilities is a guarantee of comprehensiveness in the world of uncertainty.