Problem 2
Question
Unter der Laplace-Annahme ist die Wahrscheinlichkeit für die Summe 4 beim Wurf mit 3 Würfeln () \(\frac{1}{72}\) () \(\frac{4}{216}\) () \(\frac{5}{36}\).
Step-by-Step Solution
Verified Answer
The probability is \(\frac{1}{72}\).
1Step 1 - Understand the problem
We need to calculate the probability of getting a sum of 4 when rolling 3 dice. There are three options given for the probability, and we need to determine which one is correct.
2Step 2 - Determine the number of favorable outcomes
List all possible combinations of dice rolls that add up to 4. The possible combinations (dice values) are: - (1,1,2)- (1,2,1)- (2,1,1)- (1,1,1) is not possible as the sum is 3.So, the favorable outcomes are (1,1,2), (1,2,1), and (2,1,1). Thus, there are 3 favorable outcomes.
3Step 3 - Determine the total number of possible outcomes
Each die has 6 faces. Therefore, when rolling 3 dice, the total number of possible outcomes is calculated as: \[6^3 = 216\]
4Step 4 - Calculate the probability
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. Using the results from Steps 2 and 3:\[\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{3}{216} = \frac{1}{72}\]
5Step 5 - Compare with given options
The calculated probability is \(\frac{1}{72}\). This matches the first option provided in the problem.
Key Concepts
Laplace assumptionFavorable outcomesTotal possible outcomesProbability calculation
Laplace assumption
The Laplace assumption is a fundamental concept in probability theory. It assumes that all outcomes of a random experiment are equally likely. By adopting this assumption, we simplify calculations since the probability of an event happening can be determined by dividing the number of favorable outcomes by the total number of possible outcomes.
For example, when rolling a single die, each face (1 through 6) has an equal chance of landing face up. Therefore, the probability of rolling a specific number, such as 3, is \(\frac{1}{6}\).
Similarly, in exercises like the one we have, where we roll multiple dice, if we assume the Laplace assumption, it simplifies our probability calculations and contributes to deriving an accurate result.
For example, when rolling a single die, each face (1 through 6) has an equal chance of landing face up. Therefore, the probability of rolling a specific number, such as 3, is \(\frac{1}{6}\).
Similarly, in exercises like the one we have, where we roll multiple dice, if we assume the Laplace assumption, it simplifies our probability calculations and contributes to deriving an accurate result.
Favorable outcomes
Favorable outcomes are the specific results of a random experiment that satisfy the condition we are interested in. In our exercise, we want to know how many possible combinations from 3 dice rolls add up to the sum of 4.
Here's how we calculate this:
Understanding which outcomes are favorable is crucial since it directly impacts the probability calculation.
Here's how we calculate this:
- Roll (1, 1, 2)
- Roll (1, 2, 1)
- Roll (2, 1, 1)
Understanding which outcomes are favorable is crucial since it directly impacts the probability calculation.
Total possible outcomes
Total possible outcomes refer to all the different ways an experiment can result. For our exercise, it means all different results from rolling three dice.
Each die has 6 faces, so for three dice, we calculate the total number of outcomes by multiplying the number of faces on each die:
Each die has 6 faces, so for three dice, we calculate the total number of outcomes by multiplying the number of faces on each die:
- 6 faces x 6 faces x 6 faces = 6^3 = 216
Probability calculation
Calculating probability involves determining the ratio of favorable outcomes to the total number of possible outcomes. This ratio gives us a measure of how likely an event is to occur.
The probability formula according to the Laplace assumption is:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
Applying this formula to our exercise:
\[ \text{Probability} = \frac{3}{216} = \frac{1}{72} \]
This matches with one of the given options, confirming our calculations.
The probability formula according to the Laplace assumption is:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
Applying this formula to our exercise:
- Number of favorable outcomes = 3 (from (1, 1, 2), (1, 2, 1), (2, 1, 1))
- Total number of possible outcomes = 216
\[ \text{Probability} = \frac{3}{216} = \frac{1}{72} \]
This matches with one of the given options, confirming our calculations.