Problem 3 - Algebra and Trigonometry

Question

A die is rolled. Find the probability of the given event. (a) The number showing is a six. (b) The number showing is an even number. (c) The number showing is greater than 5.

Step-by-Step Solution

Answer

(a) 1/6, (b) 1/2, (c) 1/6.

Step 1: Understanding the Event

A die has 6 faces, each showing one of the numbers from 1 to 6. We need to find the probability of rolling specific numbers on this die.

Step 2: Calculating Probability for Event (a)

For event (a), we want the number showing to be a six. There is only one favorable outcome (rolling a 6) out of 6 possible outcomes. Therefore, the probability, \( P(A) \), is given by the formula: \( P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{1}{6} \).

Step 3: Finding Even Numbers

To find the probability of rolling an even number, we identify the even numbers on a die: 2, 4, and 6. This gives us 3 favorable outcomes.

Step 4: Calculating Probability for Event (b)

The probability that the number showing is an even number is calculated by dividing the number of even numbers by the total number of outcomes. Therefore, \( P(B) = \frac{3}{6} = \frac{1}{2} \).

Step 5: Identifying Numbers Greater Than 5

For event (c), we need numbers greater than 5, which is only 6. Thus, there is 1 favorable outcome that satisfies this condition.

Step 6: Calculating Probability for Event (c)

The probability that the number showing is greater than 5 is given by \( P(C) = \frac{1}{6} \).

Key Concepts

Dice Probability

Probability is a fascinating concept that allows us to measure how likely an event is to occur. When we talk about dice probability, we're focusing on events related to rolling a die. A standard die has 6 faces, each uniquely numbered from 1 to 6. This is important because it lays the foundation for calculating probability. When we roll the die, any of these numbers could land face-up with equal likelihood.

Probability is calculated using the formula:

• \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \).

Since a die has 6 faces, the total number of possible outcomes is always 6 when you roll once. Understanding this formula is pivotal, as it applies broadly across different scenarios involving the die rolling.

Favorable Outcomes

In probability, a favorable outcome is any result that satisfies the conditions of our specific event. Let's break this down in terms of dice rolling. Suppose we want to know the probability that the die shows a six.

Out of all possible die faces, only one face has the number six, so there is 1 favorable outcome. In any probability question, identifying the correct number of favorable outcomes is crucial.

Let's recap with an example:

• If we're asked about the probability of rolling a number greater than 5, our favorable outcomes are those numbers, in this case, just the number 6 itself.

Clearly identifying the conditions and then counting the matching outcomes will lead you to find the correct probability efficiently.

Even Numbers

Even numbers on a die are easy to identify. These are numbers that can be divided by 2 with no remainder. On a six-sided die, the even numbers are 2, 4, and 6. Thus, out of 6 possible outcomes, 3 of them are even numbers.

To calculate probability for an even number event:

• Determine the favorable outcomes, which are 3 here (2, 4, and 6).
• Apply the probability formula: \( P(\text{Even Number}) = \frac{3}{6} = \frac{1}{2} \).

As you see, evenly spread outcomes (like 2, 4, and 6 on a die) make probability calculation straightforward. Recognizing patterns in numbers can simplify these problems greatly.

Greater Than 5

Determining outcomes where a number is greater than 5 on a die is simple. Since numbers on a standard die range from 1 to 6, only the number 6 is greater than 5.

For this scenario:

• The only favorable outcome is 6, making the count 1.
• Use the probability formula: \( P(\text{Greater Than 5}) = \frac{1}{6} \).

With limited choices over a small range of numbers, the exercise is a great way to practice these fundamental probability calculations. This concept of comparison (greater than or less than) reinforces numerical understanding essential in probability tasks.