If a single die is tossed twice, then what is the...

If a single die is tossed twice, then what is the probability of getting
a) a 1 followed by a 2?
b) a sum of 3?
c) a 6 on the second toss?
d) no more than two 5’s?
e) an even number followed by an odd number?

Key Concepts

Counting Principles

Counting principles involve methods for determining the total number of outcomes and the number of favorable outcomes in an experiment. Techniques from combinatorics help in efficiently counting outcomes for problems that involve sums, specific sequences, or restrictions, which is particularly useful in probability calculations.

Compound Events and the Multiplication Rule

Compound events involve the occurrence of two or more events together. For independent events like consecutive die tosses, the multiplication rule states that the probability of all events occurring is the product of their individual probabilities. This concept is used when calculating the likelihood of specific ordered outcomes across multiple events.

Independent Events

Independent events are those in which the outcome of one event does not affect the outcome of another. Each toss of a die is independent, meaning the result from one throw doesn't influence the next. This property allows for the use of the multiplication rule when determining the probability of compound events.

Sample Space

The sample space is the complete set of all possible outcomes of an experiment. In the context of rolling a die twice, the sample space consists of ordered pairs representing the result of each toss. Knowing the sample space is essential because probability is computed as the ratio of favorable outcomes to the total number of outcomes.

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