Problem 59
Question
An unbiased die with faces marked \(1,2,3,4,5\) and 6 is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is (A) \(\frac{16}{81}\) (B) \(\frac{1}{81}\) (C) \(\frac{80}{81}\) (D) \(\frac{65}{81}\)
Step-by-Step Solution
Verified Answer
The probability is \(\frac{16}{81}\).
1Step 1: Understanding the Problem
We need to find the probability that when a die is rolled four times, the minimum face value is at least 2, and the maximum face value is at most 5.
2Step 2: Identifying Favorable Outcomes
The face values of the dice that we are interested in range from 2 to 5 (inclusive). Thus, we are only considering the face values 2, 3, 4, and 5 for each roll.
3Step 3: Calculating the Total Possible Outcomes
Each roll of the dice can result in any one of 6 faces. Rolling the die 4 times independently gives a total number of outcomes equal to \[6^4 = 1296.\]
4Step 4: Calculating the Favorable Outcomes
For the condition where the minimum value is 2 and the maximum value is 5, each roll can independently result in 2, 3, 4, or 5. This gives us\[4^4 = 256\]favorable outcomes since there are 4 options (2, 3, 4, 5) for each of the 4 rolls.
5Step 5: Calculating the Probability
The probability of satisfying the condition is the ratio of the number of favorable outcomes to the total number of possible outcomes. This is computed as\[\frac{256}{1296}.\]
6Step 6: Simplifying the Probability
The fraction \(\frac{256}{1296}\) simplifies to \(\frac{16}{81}\). Thus, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is \(\frac{16}{81}\).
Key Concepts
Minimum and Maximum ValuesFavorable OutcomesTotal Possible Outcomes
Minimum and Maximum Values
When we roll a die, each face displays a specific number, ranging from 1 to 6. In probability problems, understanding the concept of minimum and maximum values can greatly help in determining the outcomes of an event. For this exercise, we focus on two key conditions regarding these values:
- **Minimum Face Value:** This is the lowest number that appears in a series of rolls. In the problem, we want the minimum value to be at least 2, meaning no roll should show a 1.
- **Maximum Face Value:** Conversely, the maximum value is the highest number that appears in any roll. In this scenario, we desire a maximum value of no greater than 5, meaning a 6 should not appear in any of the rolls.
By adhering to these conditions, we narrow down the dice outcomes that fit the criteria of having numbers strictly from 2 to 5 only.
- **Minimum Face Value:** This is the lowest number that appears in a series of rolls. In the problem, we want the minimum value to be at least 2, meaning no roll should show a 1.
- **Maximum Face Value:** Conversely, the maximum value is the highest number that appears in any roll. In this scenario, we desire a maximum value of no greater than 5, meaning a 6 should not appear in any of the rolls.
By adhering to these conditions, we narrow down the dice outcomes that fit the criteria of having numbers strictly from 2 to 5 only.
Favorable Outcomes
Favorable outcomes are those that meet all the conditions set by the problem. For our dice problem, favorable outcomes occur when each roll produces a number within our specified range of 2 to 5.
- For each independent roll of a die, there are originally 6 possible outcomes (the numbers 1 to 6).
- By excluding 1 and 6, only the numbers 2, 3, 4, and 5 remain as desirable results.
When rolling the die four times, each successful roll must exclusively yield a 2, 3, 4, or 5 to be considered favorable. This leads to favorable outcomes being calculated by raising the number of desirable results each roll to the power of the number of rolls: \[4^4 = 256\]
The determination of favorable outcomes directly influences the calculation of the problem's probability, as we'll see further.
- For each independent roll of a die, there are originally 6 possible outcomes (the numbers 1 to 6).
- By excluding 1 and 6, only the numbers 2, 3, 4, and 5 remain as desirable results.
When rolling the die four times, each successful roll must exclusively yield a 2, 3, 4, or 5 to be considered favorable. This leads to favorable outcomes being calculated by raising the number of desirable results each roll to the power of the number of rolls: \[4^4 = 256\]
The determination of favorable outcomes directly influences the calculation of the problem's probability, as we'll see further.
Total Possible Outcomes
Total possible outcomes account for every single way the dice can land irrespective of the conditions applied. In the world of probability, identifying this comprehensive set of outcomes is crucial as it acts as the denominator in probability calculations.
- With each roll, there are 6 faces the die can land on, yielding 6 outcomes.
- When the die is rolled 4 times, the possibilities magnify, as each of the 4 rolls is an independent event.
This results in the total number of possible outcomes being \[6^4 = 1296\]
These total outcomes include all combinations, with and without conditions like minimums and maximums. Only after we reduce this by examining favorable outcomes can we determine the exact probability for our desired dice event.
- With each roll, there are 6 faces the die can land on, yielding 6 outcomes.
- When the die is rolled 4 times, the possibilities magnify, as each of the 4 rolls is an independent event.
This results in the total number of possible outcomes being \[6^4 = 1296\]
These total outcomes include all combinations, with and without conditions like minimums and maximums. Only after we reduce this by examining favorable outcomes can we determine the exact probability for our desired dice event.
Other exercises in this chapter
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