Problem 14 - College Algebra

Question

The sample space of equally likely outcomes is \(\\{1,2,3,4,5,6\\} .\) Find the probability of getting: a number greater than 7 .

Step-by-Step Solution

Answer

The probability of getting a number greater than 7 is 0.

Step 1: Identifying the Sample Space

The sample space of equally likely outcomes given is \(\{1,2,3,4,5,6\}\). This means the outcomes when a die is rolled. So there are 6 possible outcomes.

Step 2: Identifying event

The event is getting a number greater than 7. However, the highest number in our sample space is 6. Therefore, there are no outcomes in our sample space that would satisfy the event.

Step 3: Calculating the Probability of the Event

The probability of an event is defined as 'the number of ways the event can occur' over 'the total number of outcomes'. In this case, the number of ways the event can occur is 0 (since it's an impossible event), and the total number of outcomes is 6. Therefore, the probability of the event is \(\frac{0}{6}=0\).

Key Concepts

Sample Space

The concept of the sample space is foundational in understanding probability. The sample space, often denoted as 'S', comprises all the possible outcomes of a particular random experiment or event. For instance, when you roll a fair six-sided die, the sample space is the set \( S = \{1,2,3,4,5,6\} \), representing each face of the die.

In a well-defined sample space, every possible outcome should be listed and each outcome should be unique - no repeats. It's critical in calculations of probability because it sets the stage for determining how likely certain events are. Depending on the context, sample spaces can be discrete, with countable outcomes, like in the roll of a die, or continuous, such as the possible values of temperature in a day.

Understanding the structure and nature of the sample space is necessary before we can define events and calculate probabilities because it directly influences the results.

Events in Probability

In probability theory, 'events' refer to sets of outcomes from the sample space that possess a particular property we are interested in. For example, an event might be 'rolling an even number' or 'drawing a red card from a deck'.

Events can range from simple, involving only a single outcome, to compound, involving multiple outcomes. It's vital to distinguish between possible events, like getting a number less than 7 when rolling a dice, and impossible events, like getting a number greater than 6 in the same scenario.

Furthermore, events can be classified as 'mutually exclusive' if they cannot occur simultaneously, or 'non-mutually exclusive' if they can. An in-depth comprehension of events is crucial for accurately calculating probabilities and understanding the likelihood of various scenarios.

Calculating Probability

Calculating probability is often seen as the heart of probability theory. It is a measure of the likelihood that a particular event will occur. To calculate the probability of an event 'E', you divide the number of ways 'E' can occur by the total number of outcomes in the sample space.

Mathematically, this is expressed as \( P(E) = \frac{\text{Number of favorable outcomes for E}}{\text{Total number of outcomes in sample space S}} \). For the die example, if we wanted to calculate the probability of rolling a number less than 3, we would see that there are two favorable outcomes (1 and 2), and so the probability would be \( P(E) = \frac{2}{6} \). If E is an impossible event, as in the case of rolling a number greater than 7 on a standard die, the probability is simply 0 because there are no favorable outcomes.

In conclusion, the probability is a fraction that lies between 0 and 1, inclusive, where 0 indicates impossibility and 1 indicates certainty. Through practice, students can become adept at identifying events and calculating their probabilities, enhancing their overall mathematical acuity.