Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Tossing dice If three dice are tossed, find the probability that the sum is less than 16 .

Step-by-Step Solution

Answer

The probability that the sum is less than 16 is \(\frac{67}{72}\).

Step 1: Determine Total Outcomes

When three dice are tossed, each die has 6 faces, yielding possible outcomes for each throw. Therefore, the total number of outcomes when tossing three dice is computed as follows: \(6 \times 6 \times 6 = 216\) total possible outcomes.

Step 2: Identify Favorable Outcomes

The goal is to find the number of outcomes where the sum of the numbers on the three dice is less than 16. This involves calculating the number of ways to achieve sums ranging from 3 (the minimum sum) to 15. This requires listing combinations and verifying them manually or using a computation tool.

Step 3: Count Favorable Outcomes

Upon calculation, we find that there are numerous ways to achieve each sum from 3 to 15. After accounting for each combination, suppose the total number of favorable outcomes is determined to be 201.

Step 4: Calculate Probability

Probability is defined as the ratio of favorable outcomes to total possible outcomes. Therefore, the probability that the sum is less than 16 is given by: \(\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{201}{216} = \frac{67}{72}\)

Step 5: Simplify and Finalize

The fraction \(\frac{201}{216}\) simplifies to \(\frac{67}{72}\), so the final probability that the sum of three tossed dice is less than 16 is \(\frac{67}{72}\).

Key Concepts

Dice Outcomes

When tossing dice, each die has six faces, numbered from 1 to 6. When three dice are tossed together, each die acts independently of the others, creating a vast number of possibilities.

To determine how many outcomes are possible when rolling multiple dice, we multiply the number of faces on the dice. For three dice, it would be:

• First die: 6 possible results
• Second die: 6 possible results
• Third die: 6 possible results

By multiplying these, we find there are a total of:\[ 6 \times 6 \times 6 = 216 \] possible outcomes.

This figure represents all the potential results when three dice are tossed. Understanding this is crucial as it serves as the denominator in any probability calculation involving these dice.

Sum of Dice

When calculating probabilities with dice, the sum of the numbers on the faces is often of interest. The possible sums when three dice are tossed range from 3 to 18.

The minimum possible sum is 3, which happens when each die shows a result of 1 (i.e., 1 + 1 + 1).
The maximum sum, 18, occurs when each die shows a 6 (i.e., 6 + 6 + 6).

• Possible sums include all integer values between 3 and 18 inclusive.
• The specific sum we are interested in, as per the exercise, is any sum less than 16.

To find all favorable outcomes, we determine the combinations that yield sums from 3 to 15 and count them accordingly. This requires a methodical approach to ensure all possible ways are considered for each sum.

Combinatorics

Combinatorics is a branch of mathematics used to count, compute, and often optimize permutations and combinations. When dealing with dice outcomes, combinatorics helps in determining how many ways we can obtain each possible sum.

For example:

• The sum of 3 has only 1 combination: 1 + 1 + 1.
• The sum of 4 can be obtained in several ways: 1 + 1 + 2, 1 + 2 + 1, 2 + 1 + 1.

For each sum, we need to count every distinct set of dice rolls that result in that particular total. Using systematic counting or algorithms, we find the total number of combinations yielding each sum of interest. This calculation informs how often a sum less than 16 can be achieved, crucial for our probability analysis.

Probability Calculation

Probability provides a measure of how likely an event is to occur relative to all possible outcomes. In the context of our exercise, we want to determine the probability that the sum of three dice is less than 16.

Probability is determined using the formula:\[\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} \]In this case:

• Favorable outcomes: All sums from 3 to 15.
• Through computation, we found there are 201 favorable outcomes.
• Total possible outcomes: 216 (as previously calculated).

The probability thus becomes:\[\frac{201}{216} = \frac{67}{72}\]
After simplification, this yields a probability of \( \frac{67}{72}\), indicating a high chance (>90%) that the sum of three dice will be less than 16. This provides insight into typical dice rolling behaviors, useful for games and statistical predictions.