Problem 2 - Basic Engineering Mathematics

Question

Find the expectation of obtaining a 4 upwards with 3 throws of a fair dice.

Step-by-Step Solution

Answer

The expectation is \( \frac{1}{2} \).

Step 1: Understanding Probability for One Throw

The probability of rolling a 4 on a fair six-sided dice is computed by recognizing there is only one outcome of a 4 in 6 possible outcomes. Thus, the probability is \( P(4) = \frac{1}{6} \). Similarly, the probability of NOT rolling a 4 is \( 1 - \frac{1}{6} = \frac{5}{6} \).

Step 2: Modeling the Problem with a Binomial Distribution

When a dice is thrown 3 times, the number of successful 4s follows a binomial distribution because the outcome of each throw is independent. The number of trials, \( n \), is 3, and the probability of success, \( p \), is \( \frac{1}{6} \). Thus, the random variable \( X \) representing throwing a 4 is binomially distributed: \( X \sim \text{Binomial}(3, \frac{1}{6}) \).

Step 3: Finding the Expectation of a Binomial Distribution

The expectation (or mean) of a binomial distribution \( X \sim \text{Binomial}(n, p) \) is given by \( E(X) = np \). For our problem, substitute \( n = 3 \) and \( p = \frac{1}{6} \), thus obtaining: \( E(X) = 3 \times \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \).

Step 4: Interpreting the Expectation Result

The expectation tells us the average number of times a 4 shows up in 3 throws of the dice. Since \( E(X) = \frac{1}{2} \), this means that, on average, a 4 will appear half a time when the dice is thrown 3 times.

Key Concepts

Probability Theory

In probability theory, we're interested in predicting how likely events are to occur. When throwing a fair six-sided dice, as each face has an equal chance of appearing, we can calculate the probability of rolling any specific number. For example, the probability of rolling a 4 is \( P(4) = \frac{1}{6} \). This is because there's only one favorable outcome (rolling a 4) among the six possible outcomes (numbers 1 through 6).

Understanding probability helps lay the foundation for more complex statistical concepts. It's crucial to note that on a fair dice, each event, like rolling a 4, is considered equally probable and independent from one another. This means that the outcome of one dice throw does not affect the outcome of another throw. Each roll of the dice is a fresh opportunity for a 4, reiterating the core concept of independent events within probability theory.

Expectation in Statistics

Expectation, also known as expected value or mean, is a fundamental concept in statistics. It provides a measure of the 'central tendency' or average outcome we might anticipate from repeated trials of a random process. In the context of a binomial distribution, such as rolling a dice multiple times, the expectation can be calculated as \( E(X) = np \), where \( n \) is the number of trials, and \( p \) is the probability of success.

For rolling a 4 in three dice throws, the expectation would be \( 3 \times \frac{1}{6} = \frac{1}{2} \). This tells us that on average, if you were to throw the dice three times many times over, you'd expect to get one 4 approximately every two sets of three throws. Though you can't get 'half' a 4 in reality, expectation is a theoretical average that provides insights into the pattern of occurrences over numerous trials.

Independent Events

Independent events are a key concept in probability and statistics. Two events are independent if the occurrence of one does not affect the probability of the other occurring. In the context of dice throwing, each throw is an independent event. This means the outcome of each roll is not influenced by the results of the previous rolls.

For example, whether you rolled a 4 on the first throw has no bearing on what you might roll on the second or third throw. This property of independence is what allows us to use models like the binomial distribution when analyzing outcomes over several trials.

• Each dice throw is considered separate.
• The probability remains constant across each throw.
• This independence simplifies the computation of probabilities over multiple trials.

Understanding independent events is crucial for correctly applying concepts like binomial distribution, where multiple independent trials contribute to statistical analysis.