Math Problem Statement

Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. Rolling two 1s followed by one 2 on three tosses of a fair die. Question content area bottom Part 1 Choose the correct answer below. ​(Type an integer or a simplified​ fraction.)

Solution

To determine if these events are independent or dependent, let's analyze them:

Independence

Two events are independent if the outcome of one event does not affect the outcome of the other. Rolling a die is a classic example of an independent event because each roll is separate and does not influence the outcome of the others. Therefore, each roll in this scenario is independent.

Combined Probability

For independent events, the probability of combined events happening in a specific sequence is the product of the individual probabilities of each event.

  1. Probability of rolling a 1 on the first roll: Since a die has 6 faces, the probability of rolling a 1 is: P(1 on first roll)=16P(\text{1 on first roll}) = \frac{1}{6}

  2. **Probability of rolling

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Math Problem Analysis

Mathematical Concepts

Probability
Independent Events

Formulas

P(A and B and C) = P(A) * P(B) * P(C) for independent events

Theorems

Multiplication Rule for Independent Events

Suitable Grade Level

Grades 9-10

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