You are given two fair dice to roll in an experiment. a)...

You are given two fair dice to roll in an experiment. a) Your first task is to report the numbers you observe. (i) What is the sample space of your experiment? (ii) What is the probability that the two numbers are the same? (iii) What is the probability that the two numbers differ by $2 ?$ (iv) What is the probability that the two numbers are not the same? b) In a second stage, your task is to report the sum of the numbers that appear. (i) What is the probability that the sum is $1 ?$ (ii) What is the probability that the sum is $9 ?$ (iii) What is the probability that the sum is $8 ?$ $13 ?$ (iv) What is the probability that the sum is

You are given two fair dice to roll in an experiment.
a) Your first task is to report the numbers you observe.
(i) What is the sample space of your experiment?
(ii) What is the probability that the two numbers are the same?
(iii) What is the probability that the two numbers differ by $2 ?$
(iv) What is the probability that the two numbers are not the same?
b) In a second stage, your task is to report the sum of the numbers that appear.
(i) What is the probability that the sum is $1 ?$
(ii) What is the probability that the sum is $9 ?$
(iii) What is the probability that the sum is $8 ?$
$13 ?$
(iv) What is the probability that the sum is

Key Concepts

Summing Outcomes

When an experiment involves combining numerical outcomes, such as adding together numbers from two random variables, the resulting values represent a new random variable. Understanding how to calculate the probabilities for sums involves identifying all pairs of outcomes that produce a particular value, which often requires systematic counting and analysis of the sample space.

Event Probability Calculation

Calculating the probability of an event involves identifying all outcomes in the event and then finding the ratio of these favorable outcomes over the total outcomes in the sample space. This concept is a cornerstone in probability theory and applies to various types of events, whether they involve matching numbers, differences, or sums.

Independent Events

Independence in probability means that the outcome of one event does not affect the outcome of another. This concept is fundamental when analyzing experiments like dice rolls, where the result of one die does not influence the result of the other, allowing probabilities to be calculated using the product of individual probabilities.

Equally Likely Outcomes

When outcomes in the sample space are equally likely, every outcome has the same probability of occurring. This assumption simplifies the calculation of probabilities since it allows one to determine the probability of an event by dividing the number of favorable outcomes by the total number of outcomes in the sample space.

Sample Space

The sample space is the set of all possible outcomes of an experiment. In probability theory, understanding the sample space is crucial because it defines the total number of outcomes from which probabilities are calculated. This concept is especially important in experiments with discrete outcomes, such as rolling dice, where each outcome is distinct and countable.

Counting Techniques

Counting techniques in combinatorics, such as listing outcomes systematically or using formulas, help in determining the number of ways events can occur. This is essential in probability calculations because it provides a means to count both the overall number of outcomes and the number of favorable outcomes for a given event.

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