Problem 10 - Algebra and Trigonometry

Question

\(9-10\) A die is rolled. Find the probability of the given event. (a) The number showing is a two or a three. (b) The number showing is an odd number. (c) The number showing is a number divisible by \(3 .\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{3}\); (b) \(\frac{1}{2}\); (c) \(\frac{1}{3}\).

Step 1: Identify Possible Outcomes

A standard die has six faces with numbers 1 through 6. Thus, the possible outcomes when rolling a die are \( \{1, 2, 3, 4, 5, 6\} \).

Step 2: Determine the Favorable Outcomes for Event (a)

For event (a), we want the die to show either a 2 or a 3. Thus, the favorable outcomes are \( \{2, 3\} \).

Step 3: Calculate the Probability for Event (a)

The probability of a specific event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Thus, for event (a), the probability \( P(A) \) is \( \frac{2}{6} = \frac{1}{3} \).

Step 4: Determine the Favorable Outcomes for Event (b)

For event (b), we want the die to show an odd number. The odd numbers between 1 and 6 are \( \{1, 3, 5\} \).

Step 5: Calculate the Probability for Event (b)

There are 3 favorable outcomes for event (b), and 6 possible outcomes in total for the die roll. Hence, the probability \( P(B) \) is \( \frac{3}{6} = \frac{1}{2} \).

Step 6: Determine the Favorable Outcomes for Event (c)

For event (c), we are looking for numbers that are divisible by 3. These numbers are \( \{3, 6\} \).

Step 7: Calculate the Probability for Event (c)

Considering the total outcomes and favorable outcomes for event (c), the probability \( P(C) \) is \( \frac{2}{6} = \frac{1}{3} \).

Key Concepts

Favorable Outcomes

In probability theory, understanding favorable outcomes is key to solving many problems. When you roll a die, each face represents a possible outcome. But, when you're interested in a specific event, you need to consider only the outcomes that meet the criteria of that event. These specific outcomes are known as the favorable outcomes. For example, if you want the die to show either a 2 or a 3, like in event (a) of our exercise, then your favorable outcomes are 2 and 3.

• Favorable outcomes for event (a): \( \ \{2, 3\} \ \)
• Favorable outcomes for event (b): odd numbers \( \ \{1, 3, 5\} \ \)
• Favorable outcomes for event (c): numbers divisible by 3 \( \ \{3, 6\} \ \)

Recognizing favorable outcomes allows you to calculate the likelihood of an event occurring.

Sample Space

In any probability problem, defining the sample space is an essential first step. The sample space is the set of all possible outcomes of an experiment or event. When rolling a standard six-sided die, the sample space includes the numbers 1 through 6.In simple terms, it's like having a list of everything that could possibly happen. For our exercise, this list is:

• Possible outcomes: \( \ \{1, 2, 3, 4, 5, 6\} \ \)

Remember, the probabilities we calculate depend on the total number of these possible outcomes in the sample space. No matter which event you are considering (a, b, or c), the sample space remains the same. This concept is crucial because it helps in calculating probabilities by providing the denominator in our fraction form for probability calculation.

Probability Calculation

Calculating probability involves a straightforward formula: \[ P( ext{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]This formula allows you to determine the likelihood of an event happening. Let's apply this to our exercise:

• For event (a), showing a 2 or 3: there are 2 favorable outcomes. With 6 possible outcomes overall, we calculate it as \( \frac{2}{6} = \frac{1}{3} \).
• For event (b), showing an odd number: there are 3 favorable outcomes. With 6 possible outcomes, the probability is \( \frac{3}{6} = \frac{1}{2} \).
• For event (c), showing a number divisible by 3: there are 2 favorable outcomes. Among 6 possible outcomes, the calculation results in \( \frac{2}{6} = \frac{1}{3} \).

This calculation method is key for determining how likely an event is to occur. It is these probability calculations that make probability theory both fascinating and useful in predicting future outcomes based on past events.