Problem 19
Question
Work each problem. A woman invites 10 relatives to a party: her mother, 2 uncles, 3 brothers, and 4 cousins. If any one guest is as likely to arrive first as any other, find each probability. A. The first guest is an uncle or a brother. B. The first guest is a brother or a cousin. C. The first guest is a brother or her mother.
Step-by-Step Solution
Verified Answer
A: \( \frac{1}{2} \), B: \( \frac{7}{10} \), C: \( \frac{2}{5} \).
1Step 1: Identify Total Number of Guests
The total number of guests the woman invites is given as 10. This includes her mother, 2 uncles, 3 brothers, and 4 cousins.
2Step 2: Calculate Probability for Part A
To find the probability that the first guest is either an uncle or a brother, count the number of uncles and brothers. There are 2 uncles and 3 brothers, making a total of 5. The probability is then the number of favorable outcomes divided by the total outcomes. \[ P(\text{uncle or brother}) = \frac{\text{number of uncles and brothers}}{\text{total number of guests}} = \frac{5}{10} = \frac{1}{2} \]
3Step 3: Calculate Probability for Part B
For this, count the number of brothers and cousins. There are 3 brothers and 4 cousins, making a total of 7. The probability is: \[ P(\text{brother or cousin}) = \frac{\text{number of brothers and cousins}}{\text{total number of guests}} = \frac{7}{10} \]
4Step 4: Calculate Probability for Part C
Here, we count the number of brothers and her mother. There are 3 brothers and 1 mother, making a total of 4. The probability is: \[ P(\text{brother or mother}) = \frac{\text{number of brothers and mother}}{\text{total number of guests}} = \frac{4}{10} = \frac{2}{5} \]
Key Concepts
Counting PrincipleSample SpaceFavorable Outcomes
Counting Principle
The counting principle is a fundamental concept in probability that helps us determine the number of possible outcomes quickly and effectively. It states that if we have multiple events, and each event has a separate number of outcomes, we can find the total number of outcomes by multiplying the number of options for each event. This is extremely useful when dealing with situations that require comprehensive listing of potential results.
In the context of the exercise, the counting principle helps to figure out the possible ways of selecting the first guest to arrive at the party. The woman has invited 10 guests, each independent in their potential to arrive first. Therefore, there are 10 possible outcomes for who might arrive first. This means when calculating the probability for each scenario, we must consider '10' as our base total, since there are 10 individuals who could be the first to arrive.
Utilizing the counting principle simplifies our probability calculations by clearly establishing how many total outcomes are in the scenario. It allows us to focus on counting the specific favorable outcomes for each part of the exercise.
In the context of the exercise, the counting principle helps to figure out the possible ways of selecting the first guest to arrive at the party. The woman has invited 10 guests, each independent in their potential to arrive first. Therefore, there are 10 possible outcomes for who might arrive first. This means when calculating the probability for each scenario, we must consider '10' as our base total, since there are 10 individuals who could be the first to arrive.
Utilizing the counting principle simplifies our probability calculations by clearly establishing how many total outcomes are in the scenario. It allows us to focus on counting the specific favorable outcomes for each part of the exercise.
Sample Space
A sample space is a term used in probability to describe the set of all possible outcomes of a particular experiment or event. In simple terms, it is like looking at all the different possibilities that could happen in a given situation. Each outcome within the sample space is an event that could potentially occur.
For this exercise of determining which guest arrives first at the party, the sample space includes all 10 guests. This means:
Understanding this concept helps with calculating probabilities correctly, as it sets the stage for recognizing how individual or grouped outcomes measure against the total number of possibilities listed in the sample space.
For this exercise of determining which guest arrives first at the party, the sample space includes all 10 guests. This means:
- One possible outcome is the mother arriving first.
- Another is any of the 2 uncles being first.
- Likewise, any of the 3 brothers or 4 cousins could also be the first to arrive.
Understanding this concept helps with calculating probabilities correctly, as it sets the stage for recognizing how individual or grouped outcomes measure against the total number of possibilities listed in the sample space.
Favorable Outcomes
Favorable outcomes are specific outcomes within the sample space that satisfy the condition of the probability we want to calculate. In simple terms, they are the outcomes we're actually interested in counting when determining probability.
In the given exercise, the woman’s guests include relatives of various kinds. The specific conditions—like someone being a brother, uncle, cousin, or her mother—determine what outcomes are considered favorable for each probability calculation.
For example, in Part A of the problem, the favorable outcomes are the cases where either an uncle or a brother arrives first. Counting these favorable outcomes means adding those who fit the criteria:
The concept of favorable outcomes is crucial since it directly connects to how we calculate the probability through the formula, \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \). Understanding what counts as favorable is key to solving probability problems accurately.
In the given exercise, the woman’s guests include relatives of various kinds. The specific conditions—like someone being a brother, uncle, cousin, or her mother—determine what outcomes are considered favorable for each probability calculation.
For example, in Part A of the problem, the favorable outcomes are the cases where either an uncle or a brother arrives first. Counting these favorable outcomes means adding those who fit the criteria:
- 2 uncles,
- and 3 brothers,
The concept of favorable outcomes is crucial since it directly connects to how we calculate the probability through the formula, \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \). Understanding what counts as favorable is key to solving probability problems accurately.
Other exercises in this chapter
Problem 18
Determine the positive integer values of \(n\), where the geometric sequence \(a_{n}\) satisfies the inequality. $$a_{n}>\frac{1}{10}, \text { where } a_{n}=\fr
View solution Problem 18
Decide whether each sequence is finite or infinite. $$-1,-2,-3,-4$$
View solution Problem 19
Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$(1+x)^{n} \geq 1+n x, \text { for } x>-1$$
View solution Problem 19
Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{10}=6, a_{12}=15$$
View solution