Problem 46 - Algebra 2

Question

Open-Ended Describe two events that are not mutually exclusive. Estimate the probability of both events occurring.

Step-by-Step Solution

Answer

The chosen not mutually exclusive events are 'getting an odd number' and 'getting a number higher than 2' when rolling a six-faced dice. The probability of these two events occurring at the same time (rolling a 3 or a 5) is 33%.

Step 1: Identify Two Non-Mutually Exclusive Events

The first step is to come up with examples of events that are not mutually exclusive. We could use the example 'getting an odd number' and 'getting a number higher than 2' when rolling a six-sided dice. These are not mutually exclusive events because it is possible to roll a 3 or 5, which satisfies both conditions.

Step 2: Calculate the Probability of Each Event

We need to calculate the probability of each event individually. In a six-sided dice, the probability of rolling an odd number (1, 3, or 5) is \(\frac{3}{6} = 0.5\) and the probability of rolling a number higher than 2 (3, 4, 5, or 6) is \(\frac{4}{6} = 0.67\)

Step 3: Calculate the Probability of Both Events Occurring Together

To find the probability of both events happening at the same time, we need to find the intersection of these two events, which occurs if we roll either a 3 or a 5. There are 2 favorable outcomes (3, 5) out of a total of 6 possible so the probability is \(\frac{2}{6} = 0.33\). Therefore, there is a 33% chance that both events will occur together when rolling a six-faced dice once.

Key Concepts

Non-Mutually Exclusive Events

In the world of probability, events can either be mutually exclusive or non-mutually exclusive. Understanding these terms is crucial when calculating probabilities. Non-mutually exclusive events are those that can occur at the same time.
For instance, consider rolling a six-sided dice and focusing on two specific events: rolling an odd number and rolling a number greater than two. These events are non-mutually exclusive because numbers like 3 and 5 are both odd and greater than two.
To further illustrate, it's helpful to list:

• Mutually exclusive events: No overlap, like heads or tails on a coin.
• Non-mutually exclusive events: Possible overlap, like the dice example mentioned.

Recognizing events that are non-mutually exclusive is a key factor in calculating probabilities correctly.

Six-Sided Dice

A six-sided dice, commonly used in board games, is a perfect example of a fair random device. Each side of a standard dice represents a different whole number from 1 to 6.
When you roll a dice, each number has an equal opportunity to land face up. This means that the probability of landing on any specific number is purely chance-driven and equal for each side. With a total of six sides, each number has a probability of \(\frac{1}{6}\) or approximately 16.67% to be rolled.

• Six numbers: 1, 2, 3, 4, 5, 6
• Equally probable outcomes
• Total sum of probabilities is 1 (or 100%)
Knowing these basic principles helps when you need to determine more complex probability questions involving dice.

Calculating Event Probability

Calculating the probability of events, especially those involving dice, may seem challenging but is straightforward if broken down into steps. Let's focus on how to calculate the probability of two non-mutually exclusive events using a six-sided dice example.
Firstly, determine the probability of each individual event. In our scenario, 'rolling an odd number' and 'rolling a number greater than two' are the events:

• Probability of odd numbers (1, 3, 5): \(\frac{3}{6} = 0.5\)
• Probability of numbers greater than 2 (3, 4, 5, 6): \(\frac{4}{6} \approx 0.67\)

Next, you calculate the probability of these events occurring together, their intersection. For our dice example:

• Outcomes that satisfy both: 3 or 5
• Probability of intersection: \(\frac{2}{6} \approx 0.33\)

Understanding how to find the overlap in non-mutually exclusive events simplifies the process for determining the combined probability, ensuring accuracy in your calculations.