91Ó°ÊÓ

A July 21, 2009 (just a reminder that July 21 is National Junk Food Day) survey on www. HuffingtonPost.com asked people to choose their favorite junk food from a list of choices. Of the 8002 people who responded to the survey, 2049 answered chocolate, 345 said sugary candy, 1271 mentioned ice cream, 775 indicated fast food, 650 said cookies, 1107 mentioned chips, 490 said cake, and 1315 indicated pizza. Although the results were not broken down by gender, suppose that the following table represents the results for the 8002 people who responded, assuming that there were 4801 females and 3201 males included in the survey. $$ \begin{array}{lcc} \hline \text { Favorite Junk Food } & \text { Female } & \text { Male } \\ \hline \text { Chocolate } & 1518 & 531 \\ \text { Sugary candy } & 218 & 127 \\ \text { Ice cream } & 685 & 586 \\ \text { Fast food } & 312 & 463 \\ \text { Cookies } & 431 & 219 \\ \text { Chips } & 458 & 649 \\ \text { Cake } & 387 & 103 \\ \text { Pizza } & 792 & 523 \\ \hline \end{array} $$ a. If one person is selected at random from this sample of 8002 respondents, find the probability that this person i. is a female ii. responded chips iii. responded chips given that this person is a female iv. responded chocolate given that this person is a male b. Are the events chips and cake mutually exclusive? What about the events chips and female? Why or why not? c. Are the events chips and female independent? Why or why not?

Step by step solution

01

Compute Marginal Probabilities

Start by computing the marginal probabilities for all options, we need them for later computations. We have 4801 females out of total 8002, so the probability a person is female is \( \frac{4801}{8002} = 0.59985 \). There are 1107 people who responded chips out of 8002, so the probability that a person responded chips is \( \frac{1107}{8002} = 0.13834 \).

02

Compute Conditional Probabilities

Compute the conditional probabilities: For the event 'responded chips given that this person is a female', note that when the person is a female, our sample space changes to 4801 (the number of females). There are 458 females that responded chips out of these 4801 females, so the conditional probability is \( \frac{458}{4801} = 0.09540 \). Similarly, for the event 'responded chocolate given that this person is a male', our sample space changes to 3201 (the number of males). There are 531 males that responded chocolate out of these 3201 males. Therefore, the conditional probability is \( \frac{531}{3201} = 0.16588 \).

03

Discuss Mutual Exclusion

For two events to be mutually exclusive, they cannot occur at the same time. If a respondent said chips, they could still have said cake. Therefore, chips and cake are not mutually exclusive. Similarly, a respondent could be female and have responded chips. So, chips and female are not mutually exclusive.

04

Discuss Independence

Two events are independent if the occurrence of one does not affect the occurrence of the other. Therefore, if chips and female are independent, the probability of chips given female should equal the probability of chips - which is not the case here (0.09540 ≠ 0.13834). Therefore, chips and female are not independent events.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Understanding Marginal Probability

Marginal probability is a fundamental concept in statistics that helps us understand the likelihood of a single event occurring within a set of possibilities. It is calculated by dividing the number of successful outcomes for that event by the total number of possible outcomes. This gives us a probability value between 0 and 1.
In the survey of favorite junk foods, we calculated the marginal probability that a respondent is female by dividing the number of females (4801) by the total number of respondents (8002). Thus, the marginal probability of choosing a female respondent is \( \frac{4801}{8002} \approx 0.600 \).
Similarly, to find the probability that a respondent chose chips as their favorite junk food, we take the number of people who chose chips (1107) divided by the total respondents, resulting in a marginal probability of \( \frac{1107}{8002} \approx 0.138 \).
These calculations are integral to further probability analysis and help us establish a baseline for comparing other probabilities.

Delving into Conditional Probability

Conditional probability is used when we want to find the likelihood of an event occurring given that another event has already occurred. This type of probability narrows down the sample space to only those situations where the given event is true.
For instance, to find the probability that a respondent said chips given that the respondent is female, we focus solely on the female respondents. Out of 4801 females, 458 chose chips, so the conditional probability is \( \frac{458}{4801} \approx 0.095 \).
Similarly, to determine the probability that a male respondent chose chocolate, we only consider the 3201 male respondents. With 531 males having chosen chocolate, this probability is \( \frac{531}{3201} \approx 0.166 \).
This form of probability is crucial for understanding how certain conditions affect the likelihood of various outcomes and is widely used across different fields, particularly in making informed decisions and predictions.

Exploring Mutual Exclusion

The concept of mutual exclusion involves determining whether two events can occur simultaneously. If two events are mutually exclusive, then the occurrence of one will prevent the other from happening at the same time.
In our survey, we were asked to check if selecting ‘chips’ and ‘cake’ as favorite junk foods are mutually exclusive events. Since respondents can list more than one favorite, it's possible for someone to prefer both chips and cake, meaning they are not mutually exclusive.
Similarly, when considering the events of a respondent being 'female' and having chosen 'chips', it is evident that these events can occur simultaneously. A female respondent can prefer chips. Thus, these events are also not mutually exclusive.
Understanding mutual exclusion helps us determine if events have overlapping possibilities, which is essential in proper probability assessment and decision-making.

Identifying Independent Events

Two events are considered independent when the occurrence of one does not affect the likelihood of the other occurring. In mathematical terms, events A and B are independent if \( P(A \cap B) = P(A) \times P(B) \).
In our exercise, we examined whether being female and choosing chips as favorite junk food are independent events. For independence, the probability of choosing chips should be the same regardless of whether the respondent is female. However, the conditional probability of choosing chips given female respondents (\( \frac{458}{4801} \approx 0.095 \)) differs from the marginal probability of choosing chips (\( \frac{1107}{8002} \approx 0.138 \)).
This difference indicates that these events are not independent, as being female seems to affect the likelihood of choosing chips. Recognizing dependent or independent events is key in probability and impacts how we interpret data and model systems.