(a) Suppose that you throw 3 dice. Find the probability that...

(a) Suppose that you throw 3 dice. Find the probability that you get at least one 2 .
(b) Suppose that you throw 2 dice 20 times. Find the probability that you get at least one $(1,1),$ that is, snake-eyes.

Key Concepts

Complementary Probability

This is a method used to simplify the calculation of probabilities when dealing with events like 'at least one occurrence.' Instead of calculating the probability directly, you calculate the probability that the event does not occur at all and subtract that value from one. This technique is particularly useful when dealing with multiple independent trials.

Independent Events

In probability theory, events are independent if the outcome of one does not affect the outcome of another. In dice throws, each throw is independent of the others, which means the probability for each throw remains constant regardless of previous outcomes. This concept is essential when calculating probabilities over multiple trials.

Bernoulli Trials

A Bernoulli trial is an experiment that results in one of two outcomes: success or failure. Repeated independent experiments, like throwing dice multiple times, can be modeled as Bernoulli trials. Calculating the probability of at least one success in a sequence of Bernoulli trials often involves using the complementary probability approach.

Counting and Sample Space

Understanding the sample space—the set of all possible outcomes—is fundamental in probability. In scenarios involving dice, determining the total number of outcomes allows for the calculation of the probability of specific events. This concept helps in establishing the basis for further probability computations, such as those using the complement rule.

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