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Chapter 12: Problem 100

In a single throw of a pair of dice, find the probability of obtaining a total of 4 or less.

Short Answer

Expert verified
The probability of obtaining a total of 4 or less in a single throw of a pair of dice is \(\frac{1}{6}\).

Step by step solution

01

Determine the Total Possible Outcomes

When rolling two dice, there are 6 sides on each die, so there are 6 x 6 = 36 possible combinations of outcomes for the two dice.
02

Determine the Favorable Outcomes

To find the favorable outcomes, we need to determine all possible combinations of rolls that result in a sum of 4 or less. Let's list these in a table: Die 1 | Die 2 | Sum ------|-----|---- 1 | 1 | 2 1 | 2 | 3 1 | 3 | 4 2 | 1 | 3 2 | 2 | 4 3 | 1 | 4 As we can see, there are 6 such combinations.
03

Calculate Probability

Now, we can use the formula for probability to calculate the answer: \[P(\text{sum} \le 4) = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}}\] Plugging in the numbers we found earlier, we get: \[P(\text{sum} \le 4) = \frac{6}{36}\] Simplify the fraction to get: \[P(\text{sum} \le 4) = \frac{1}{6}\] Thus, the probability of obtaining a total of 4 or less in a single throw of a pair of dice is \(\frac{1}{6}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dice Outcomes
When we roll a pair of dice, each one has 6 faces showing numbers from 1 to 6, which means there are many possible results from each roll.
Together, these dice can create various combinations.
The important thing to remember is that each die is independent, meaning the face shown on one die doesn't affect the other.
  • Each die by itself has 6 possibilities.
  • For two dice, we multiply the possibilities, resulting in 6 x 6 = 36 outcomes.
These 36 combinations represent all potential results when rolling the two dice.
From (1,1) both showing 1s to (6,6) showing 6s, each combination is unique. Understanding these possible outcomes is crucial for calculating any probability involving dice.
Favorable Outcomes
Favorable outcomes are the results that meet the condition we are interested in.
For this exercise, we're focusing on obtaining a sum of 4 or less when the two dice are rolled.
  • Let's lay out the combinations: (1,1), (1,2), (1,3), (2,1), (2,2), and (3,1).
Each of these pairs represents a total sum equal to or less than 4.
There are 6 such combinations, hence 6 favorable outcomes.
Remember, favorable outcomes depend entirely on the condition given in the problem.
So, if the requirement were different, like getting an exact sum of 3, the list would change.
Probability Calculation
Calculating probability helps determine how likely an event is to occur.
The formula is:\[P( ext{Event}) = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}}\]For this specific problem:
  • The numerator represents the 6 combinations that suit our condition.
  • The denominator is the total 36 combinations possible with two dice.
Thus, the probability is:\[P(\text{sum} \le 4) = \frac{6}{36}\]Simplifying this fraction provides us 1 out of 6, or \(\frac{1}{6}\).
Therefore, there's a 1 in 6 chance, or about 16.67%, of rolling a sum of 4 or less.
This calculation gives insight into how often this result may happen if you roll the dice many times.

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