Problem 3
Question
A die is rolled repeatedly until a 6 falls uppermost. Let the random variable \(X\) denote the number of times the die is rolled. What are the values that \(X\) may assume?
Step-by-Step Solution
Verified Answer
The random variable \(X\) may assume any positive integer value, as it represents the number of times a die is rolled until a 6 appears (\(X = 1, 2, 3, 4, ... \)). There is no upper limit to the number of times a die can be rolled until a 6 appears.
1Step 1: Identify the possible outcomes of rolling a die
There are a total of 6 sides on a die with numbers ranging from 1 to 6. So, there are 6 possible outcomes when rolling the die, with each outcome having a probability of \(\frac{1}{6}\).
2Step 2: Determine the possible values of X
Since X is the number of times a die is rolled until a 6 appears, the possible values of X can be any natural number, i.e., 1, 2, 3, 4, 5, and so on. There is no upper limit to the number of times a die can be rolled until a 6 appears.
To summarize, the values that the random variable \(X\) may assume are all positive integers (\(X = 1, 2, 3, 4, ... \)).
Key Concepts
Random VariableGeometric DistributionDiscrete Probability
Random Variable
In the study of probability, a random variable is essentially a variable that takes on different numerical values, each with a certain probability, resulting from a random experiment. These values can represent possible outcomes of an experiment such as rolling a die or drawing a card from a deck.
Random variables are typically classified into two main types: discrete and continuous. A discrete random variable is one that has a countable number of possible outcomes, like the number of times you might flip a coin before it lands heads. In contrast, a continuous random variable has an infinite number of possible outcomes within a given range, such as the exact amount of time it takes for an ice cream to melt.
The random variable in the exercise, denoted by \(X\), represents the number of times a die must be rolled before a 6 is observed. Since a die has a finite number of sides, and we're counting the number of rolls, \(X\) is decidedly a discrete random variable. Each roll is independent, meaning the result of one roll doesn't affect the next, and \(X\) assumes positive integer values starting from 1 - illustrating the quintessential characteristics of a discrete probability distribution.
Random variables are typically classified into two main types: discrete and continuous. A discrete random variable is one that has a countable number of possible outcomes, like the number of times you might flip a coin before it lands heads. In contrast, a continuous random variable has an infinite number of possible outcomes within a given range, such as the exact amount of time it takes for an ice cream to melt.
The random variable in the exercise, denoted by \(X\), represents the number of times a die must be rolled before a 6 is observed. Since a die has a finite number of sides, and we're counting the number of rolls, \(X\) is decidedly a discrete random variable. Each roll is independent, meaning the result of one roll doesn't affect the next, and \(X\) assumes positive integer values starting from 1 - illustrating the quintessential characteristics of a discrete probability distribution.
Geometric Distribution
The geometric distribution is relevant when we discuss the probability of the first occurrence of success in a sequence of independent and identically distributed (i.i.d.) Bernoulli trials. Each trial has only two outcomes: 'success' or 'failure,' and we're interested in how long it takes for a 'success' to occur. This distribution is a perfect model for our die-rolling scenario given in the exercise.
For instance, if we define getting a '6' on a die as 'success', then the random variable \(X\), which represents the number of times the die is rolled until the first '6' appears, follows the geometric distribution. The probability that the first '6' appears on the \(n\)-th trial is given by \(P(X = n) = (1-p)^{n-1} \times p\), where \(p\) is the probability of 'success' on any given trial. In the case of a fair die, \(p = \frac{1}{6}\), because this is the probability of rolling a '6'.
The defining characteristic of a geometric distribution is that it has the 'memoryless' property, meaning the probability of achieving our 'success' on the next trial does not depend on how many trials have already occurred.
For instance, if we define getting a '6' on a die as 'success', then the random variable \(X\), which represents the number of times the die is rolled until the first '6' appears, follows the geometric distribution. The probability that the first '6' appears on the \(n\)-th trial is given by \(P(X = n) = (1-p)^{n-1} \times p\), where \(p\) is the probability of 'success' on any given trial. In the case of a fair die, \(p = \frac{1}{6}\), because this is the probability of rolling a '6'.
The defining characteristic of a geometric distribution is that it has the 'memoryless' property, meaning the probability of achieving our 'success' on the next trial does not depend on how many trials have already occurred.
Discrete Probability
In probability theory, discrete probability deals with events that have distinct and separate values. This is the type of probability we calculate for random variables like our \(X\) in the die-rolling exercise, where the outcomes are a countable number of discrete values.
For a discrete random variable, we can express its behavior with a probability mass function (PMF), which maps each value of the random variable to its probability. For our die problem, we are interested in determining the PMF of rolling a '6', which is a discrete probability distribution. The PMF specifies the probability that a discrete random variable is exactly equal to some value. Over the entirety of its domain, the PMF must sum to one, reflecting the certainty that one of the possible outcomes will occur.
Importantly, when it comes to discrete probability distributions like in our example, certain formulas and characteristics apply that make calculating probabilities more systematic, as opposed to continuous distributions that usually require integrals and calculus to assess probability measures.
For a discrete random variable, we can express its behavior with a probability mass function (PMF), which maps each value of the random variable to its probability. For our die problem, we are interested in determining the PMF of rolling a '6', which is a discrete probability distribution. The PMF specifies the probability that a discrete random variable is exactly equal to some value. Over the entirety of its domain, the PMF must sum to one, reflecting the certainty that one of the possible outcomes will occur.
Importantly, when it comes to discrete probability distributions like in our example, certain formulas and characteristics apply that make calculating probabilities more systematic, as opposed to continuous distributions that usually require integrals and calculus to assess probability measures.
Other exercises in this chapter
Problem 3
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Find the expected value of a random variable \(X\) having the following probability distribution: $$\begin{array}{lcccccc}\hline \boldsymbol{x} & -5 & -1 & 0 &
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Let \(A\) and \(B\) be two events in a sample space \(S\) such that \(P(A)=.6\) and \(P(B \mid A)=.5\). Find \(P(A \cap B)\).
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The probability distribution of a random variable \(X\) is given. Compute the mean, variance, and standard deviation of \(X\). $$\begin{array}{lllllll}\hline \b
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