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Chapter 5: Problem 26

Let the sample space be \(S=\\{1,2,3,4,5,6,\) 7,8,9,10\\}\(.\) Suppose the outcomes are equally likely. Compute the probability of the event \(F="\) an odd number."

Short Answer

Expert verified
The probability of event \(F\) is \(\frac{1}{2}\).

Step by step solution

01

Identify the sample space

The sample space is given as the set of all possible outcomes. Here, the sample space is \(S=\{1,2,3,4,5,6,7,8,9,10\}\).
02

Determine the event set

The event \(F\) is defined as obtaining an odd number from the sample space. Let's list all the odd numbers in the sample space: \(F=\{1,3,5,7,9\}\).
03

Count the number of favorable outcomes

Count the number of elements in the set \(F\). The elements of \(F\) are \(1, 3, 5, 7, 9\), so the number of favorable outcomes is 5.
04

Count the total number of outcomes

Count the total number of elements in the sample space \(S\). The elements of \(S\) are \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10\), so the total number of outcomes is 10.
05

Apply the probability formula

The probability of an event occurring is given by the formula: \[P(F) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]Substitute the values: \[P(F) = \frac{5}{10} = \frac{1}{2}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Space
In probability theory, the first critical concept you need to understand is the sample space. The sample space represents all possible outcomes of an experiment. In our example, the sample space consists of the numbers 1 through 10. When we list these outcomes, we write: \[S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\]Think of the sample space as your starting point. It's like having all possible options laid out in front of you. Whether you're rolling a die, flipping a coin, or picking a number from a set, identifying the sample space is crucial.
Equally Likely Outcomes
Next, let's talk about equally likely outcomes. When we say the outcomes are equally likely, it means each outcome has the same chance of occurring. In our example, each number in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is equally likely to be selected.
  • This is like rolling a fair die where each number from 1 to 6 has an equal probability.
  • There's no bias; therefore, every outcome has an equal chance.
This assumption simplifies our calculations since we don't need to adjust for any favorability among the outcomes. All 10 outcomes in our sample space are equally probable.
Favorable Outcomes
Once you have your sample space, you need to identify the favorable outcomes. These are the outcomes that match the event you're trying to assess. For our problem, the event is getting an odd number.First, we list the odd numbers in the sample space. These odd numbers are:
  • {1, 3, 5, 7, 9}
Next, count the number of favorable outcomes. In this case, we have 5 favorable outcomes. Understanding favorable outcomes helps us connect the event we care about to our sample space. By knowing which results count as 'favorable,' we simplify the calculation of probabilities.
Probability Formula
Finally, let's use the probability formula to find the probability of getting an odd number. The formula is straightforward: \[ P(F) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]We already know the number of favorable outcomes (5) and the total number of outcomes in the sample space (10), so we substitute these values into the formula:
  • \[ P(F) = \frac{5}{10} = \frac{1}{2} \]
This tells us that the probability of drawing an odd number from our sample space is 0.5 or 50%. Using the probability formula allows us to quantify the likelihood of an event in a clear and mathematical way.

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