Probability Workshop

Analysing scenarios, identifying appropriate probability models, and applying those models to answer probability questions

�About These Slides

• Each slide contains a scenario and questions associated with events and probabilities of events
• You’ll need to decide what distribution applies
• Equally likely (uniform) outcomes
• Binomial distribution
• Normal distribution
• Then you’ll need to apply what you know about working with those distributions to answer the questions

�Commute Times

Scenario: Commute times to campus follow an approximately normal distribution with a mean of 25 minutes and a standard deviation of 5 minutes.

• What is the probability a randomly selected student’s commute takes less than 20 minutes?
• What is the probability a commute takes more than 30 minutes?
• What is the probability a commute takes between 22 and 28 minutes?
• How long are the longest 10% of commute times?
• What is the probability that a randomly sampled collection of 10 commuters reports an average commute time of at least 28 minutes? (Why can’t you answer this one yet?)

�Club Participation and GPA

Scenario: A survey of 300 students asks about participation in campus clubs and whether their GPA is above 3.5. The table below summarizes the results:

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1. What is the probability a randomly selected student has a GPA above 3.5?
2. What is the probability a randomly selected student is both a club member and has a GPA above 3.5?
3. What is the probability a student has a GPA above 3.5 given they are not a club member?

4. Are GPA and club membership independent? Justify your answer.

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�Job Interviews

Scenario: A career services center has tracked that students historically are invited to interviews for 37% of the jobs they apply to. A student applies to 18 jobs.

• What is the probability they are invited to exactly 6 interviews?
• What is the probability they are invited to at least 8 interviews?
• What is the probability they are invited to fewer than 3 interviews?

�Exam Scores

Scenario: Exam scores in a statistics class are normally distributed with a mean of 78 and a standard deviation of 8.

• What is the probability a randomly selected student scores more than 90?
• What is the probability a score is between 70 and 85?
• What is the cutoff score for the top 5% of the class?

�Spending Habits

Scenario: A marketing survey asks 200 respondents about their weekly spending at a coffee shop and categorizes them by spending range and frequency of visits. The results are summarized below:

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1. What is the probability a respondent spends $10–$20 per week?
2. What is the probability a respondent is a frequent visitor and spends > $20 per week?
3. What is the probability a respondent is a frequent visitor or spends > $20 per week?
4. What is the probability a respondent spends > $20 given they are an occasional visitor?

�Length of Salamanders

Scenario: The lengths of salamanders are nearly normally distributed with a mean of 16 cm and a standard deviation of 2.5 cm.

• What is the probability a salamander is shorter than 13.25 cm?
• What is the probability a salamander is at least 20 cm long?
• What length are the shortest 25% of salamanders?

�Online Reviews

Scenario: A consumer watchdog group notes that about 4% of customer reviews left on an online retail site are fake. For 43 randomly selected reviews:

• What is the probability exactly 3 are fake?
• What is the probability at least 5 are fake?
• What is the probability that none are fake?
• A random sample of 350 reviews from a particular seller’s account shows that 9% of reviews left were fake. What is the probability of observing at least 9% fake reviews out of a random sample of 350? What might this suggest about the particular seller?

�Example: Hospital Length of Stay

Scenario: A local hospital claims that the average length of stay for patients is 5.2 days, with a standard deviation of 2.1 days.

1. What is the probability of an individual stay being less than 3 days?
2. What is the probability of an individual stay exceeding 10 days?
3. What is the probability of an individual stay being between 3 days and 10 days?

�Psychology Experiment

Scenario: In a psychology experiment on memory, participants were shown a list of 10 words and then asked to choose pictures of those words from a grid of 25 images. In that study, it was shown that 82% of participants correctly selected all 10 images. In a study attempting to replicate these results with 100 participants:

• What is the probability exactly 67 correctly identify the image?
• What is the probability that more than 42 but at most 67 correctly identify the image?
• The results of the replication study showed that 67 participants correctly identified all images. What is the probability that at most 67 correctly identify the images? Is this cause for concern?

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�Retail Demand

Scenario: Weekly demand for a product is nearly normal with a mean of 633 units and a standard deviation of 59 units.

• What is the probability of demand exceeding 700 units?
• What is the probability that demand is between 550 and 650 units (inclusive)?
• What is the 75th percentile of demand?

�Campus Events

Scenario: A student group estimates that 30% of attendees at their events are first-time participants. Out of 23 attendees:

• What is the expected number of first-time attendees?
• What is the standard deviation for number of first-time attendees?
• Should the student group be surprised if at least 15 are first-time participants?

�Marketing Campaign

Scenario: A marketing campaign predicts a 25% response rate for their ads. If 164 people are shown the ad:

• What is the probability exactly 36 respond?
• What is the probability at least 50 respond?
• What is the probability fewer than 20 respond?
• What is the probability that at least 20 but fewer than 50 respond?

�Car Speeds in School Zones

Scenario: A traffic study from a year ago showed that the speeds of cars passing through a school zone were normally distributed with a mean of 25 mph and a standard deviation of 3 mph.

• What is the probability a car is traveling through at less than 20 mph?
• What is the probability a car is traveling through between 23 and 28 mph?
• At what speed do the fastest 5% of drivers pass through the school zone?

�Grocery Spending

Scenario: Weekly grocery spending for members at a grocery store can be approximated by a normal distribution with a mean of $143 and a standard deviation of $37.

• What is the probability that a member spends more than $190?
• What is the probability that a member spends between $100 and $150?
• The grocery store provides a “$10 off” coupon to the top 15% of spenders each week. How much does a member need to spend in order to get the coupon?

Exit Ticket

Navigate to our MAT240 Exit Ticket Form, answer the questions, and complete the task below.

Note. Today’s discussion is listed as 7. Probability Workshop

Task: How are you feeling as we head into our first exam experience in MAT240? What are you planning to do to prepare? What can I do to help you?

�Next Time…

• What we’ll be doing…
• Exam Week
• How to prepare…
• Review your class notes
• Complete the optional Practice for Exam I on MyOpenMath
• Review your optional extra practice from MyOpenMath
• Homework: Prepare for Exam I

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