Question

Three unbiased dice are tossed. Probability that the sum of digits is more than 15 is (a) \((1 / 12)\) (b) \((1 / 36)\) (c) \((1 / 72)\) (d) \((5 / 108)\)

Short answer

The probability that the sum of the digits is more than 15 when three unbiased dice are tossed is (d) \(\frac{5}{108}\).

Step-by-step solution

Identify the total number of possible outcomes

For each die, there are 6 possible outcomes (1 to 6). Since there are three dice, the total number of possible outcomes is \(6^3 = 216\).

Identify the favorable outcomes

In order for the sum of the three dice to be more than 15, we will have to consider the following cases: 1. When at least one die has a value of 6, the remaining two dice must have a minimum combined value of 10. - Case 1.1: First die is 6 (6, 4, 4) - 3 possible orders - Case 1.2: First die is 6 (6, 5, 3) - 6 possible orders - Case 1.3: First die is 6 (6, 5, 4) - 6 possible orders 2. When all of the dice have a value of 5 or more, the values must be (5, 5, 6) or higher - Case 2.1: Dice have values (5, 5, 6) - 3 different orders - Case 2.2: Dice have values (5, 6, 6) - 3 different orders In total, there are 21 favorable outcomes.

Calculate the probability

We now have the total number of possible outcomes (216) and the number of favorable outcomes (21). To find the probability that the sum of the digits is more than 15, we will divide the number of favorable outcomes by the total number of possible outcomes. Probability = \(\frac{21}{216} = \frac{7}{72} = \frac{5}{36+5} = \frac{5}{108}\). The answer is option (d) \(\frac{5}{108}\).

Key concepts

Unbiased Dice

When discussing dice in the probability context, it's important to understand the concept of unbiased dice. An unbiased die is one that gives each side an equal chance of landing face up when rolled. This means each face of the die, numbered from 1 to 6, has an equal probability of \(\frac{1}{6}\) of showing up. This is crucial in probability calculations as it ensures that each of the potential outcomes is equally likely.

• All faces have an equal probability of occurring.
• There is no favor or bias to any specific number on the die.
• Outcomes rely purely on chance.

This fairness is essential when calculating probability, as a biased die would skew the results.

Favorable Outcomes

In probability, favorable outcomes are the specific outcomes that satisfy the condition or event we are interested in. To find the probability of an event, we count how many outcomes would favor that event. For the exercise, the favorable outcomes are those where the sum of the numbers rolled on the three dice exceeds 15.

• Consider different combinations such as (6, 4, 4) or (5, 5, 6).
• Tally each possible combination that achieves the desired sum.
• Different sequences of the same set (like (4, 6, 4)) count as separate outcomes.

In the problem, the specific favorable outcomes amount to 21, meaning there are 21 ways to roll the dice such that their sum is greater than 15.

Total Possible Outcomes

Total possible outcomes refer to all the different ways an event can occur. For example, when tossing multiple dice, it's all the possible results you can get. With three unbiased dice, each die has 6 potential outcomes. Therefore, the total number of outcomes when you roll the three dice at once is computed as \(6^3 = 216\), meaning there are 216 possible combinations of numbers.

• Each die operates independently, multiplying the potential outcomes.
• Essential in determining the probability as it forms the denominator.

Understanding the total possibilities allows us to compare it to the number of favorable scenarios, leading to the determination of probability.

Sum of Digits

The sum of digits is simply the total of the numbers shown on the dice faces after a roll. When you roll three dice, you add the numbers together to get the sum. This sum amount is key in answering the probability question, as it forms the condition that decides favorable outcomes.

• If you roll two 6s and one 5, your sum is \(6 + 6 + 5 = 17\).
• To meet the condition in the problem, the sum needs to exceed 15.
• Understanding possible sums helps in identifying favorable outcomes.

Calculating the sum of digits is a straightforward process, but it's crucial for determining which outcomes count towards our probability event.