Problem 27 - Precalculus with Limits

Question

In Exercises 25 - 30, find the probability for the experiment of tossing a six-sided die twice. The sum is less than \( 11 \).

Step-by-Step Solution

Answer

The probability that the sum of two dice rolls is less than 11 is \( \frac{11}{12} \).

Step 1: Number of outcomes with Two Dice

When two six-sided dice are rolled, the number of possible outcomes is \( 6 \times 6 = 36 \) because each dice has 6 faces and the outcomes of the two dice are independent.

Step 2: Outcomes resulting in a sum less than 11

The outcomes in which the sum of numbers is less than 11 are all pairs (i,j) such that \( 1 \leq i \leq 6 \) and \( 1 \leq j \leq 6 \) excluding the pairs (5,6), (6,5) and (6,6). This is because they make the sum equal to or more than 11. Thus, the number of such outcomes is \( 36 - 3 = 33 \).

Step 3: Probability calculation

The probability is calculated by dividing the number of favorable outcomes (that is, the outcomes where the sum of the dice is less than 11) by the total number of outcomes. So, the required probability is \( P(E) = \frac{33}{36} = \frac{11}{12} \).

Key Concepts

Outcomes with Two Dice

When you toss two six-sided dice, the possible outcomes increase greatly compared to just one die. Each side of the first die can pair with any side of the second die. Therefore, if each die has 6 faces, you calculate the total possible outcomes by multiplying the number of faces on both dice: 6 for the first die and 6 for the second die. This results in a total of 36 outcomes. In essence, every combination such as "1 on the first die and 2 on the second, or 4 on the first and 5 on the second" counts as a distinct outcome. All these combinations form the fundamental basis of calculating probabilities with two dice.

Probability Calculation

Probability is about determining how likely an event is to occur. With two dice, once you know the total number of possible outcomes, you can find the probability of a specific event by identifying the favorable outcomes for that event.For example, if you want to find the probability that the sum of the two dice is less than 11, you first calculate the total outcomes leading to this event. Since we know there are 36 possible outcomes when two dice are rolled, we explore which of these combinations result in sums less than 11.After discounting these outcomes from all possibilities, the remaining favorable outcomes are 33. By using the formula for probability:\[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]This gives you the probability of the sum being less than 11 as \( \frac{33}{36} \), which can be simplified to \( \frac{11}{12} \). Thus, there's a high likelihood that the sum will be less than 11 when tossing two dice.

Sum of Dice Outcomes

The sum of outcomes is a crucial aspect when dealing with probabilities of tossing two dice. Each die can show a number between 1 and 6. The smallest possible sum is 1 + 1 = 2, and the largest is 6 + 6 = 12. Understanding the range of possible outcomes helps determine which combinations are included when looking for a specific sum. Calculating the sum involves pairing each number on the first die with every number on the second. For sums less than 11, you add each possible pair until reaching the cutoff points. Pairs like (5,6), (6,5), and (6,6) lead to sums of 11 or greater and thus are excluded when seeking sums below 11. Recognizing the possible sums helps in quickly assessing which outcomes fall under specific probability criteria, particularly in identifying trends like sums below, above, or equal to a given number.