Driving and Risk

Javier will be a high school senior next year Javier will be a high school senior next year. He wants to get a vehicle to celebrate his graduation. Javier’s mother researched vehicle safety and found that 1 of every 6 teenage drivers was involved in some kind of accident. While talking to his math teacher, Javier mentioned that he did not think the risk was high enough to be concerned. Javier decided to survey 500 students, 230 of whom were male, to help him convince his mother to allow him to get a vehicle. No student has both a car and a motorcycle. Car Motorcycle Males with vehicle 150 23 Males involved in accident 40 6 Females with vehicle 225 10 Females involved in accident 15 4

1. Draw a Venn diagram and a tree diagram of the data.

2. Using the data, what is the probability that Javier will be involved in an accident if he gets a motorcycle? Of the 23 males who had a motorcycle, 6 were involved in an accident. Therefore, the probability is 6/23 , or 26%. 3. Based on these survey data, Javier told his mother that he only has a 1% chance of getting in an accident. Is he correct? Why or why not? Javier is not correct because he is using only the probability if he gets a motorcycle. Since he is also considering a car, the total probability of getting in an accident based on Javier’s survey data is 0.08 + 0.01, which equals 0.09, or 9%. Taking a different approach, Javier could be considerably off in his comment to his mother. It could be argued that 46 of the 230 males surveyed were involved in an accident, or 20%. 4. Use your Venn diagram to write three facts that help Javier convince his mother to let him get a vehicle. • Only 46 males were involved in an accident. • Of the 500 people surveyed, 435 people did not get in an accident. • 127 males with vehicles did not have an accident.

5. What probability model would you advise Javier to use when he tries to convince his mother? Advise Javier to use the tree diagram. The tree diagram separates the males from the females. The female data helps Javier’s cause because females get in fewer accidents per person with a vehicle.

Choosing Classes

• Ms. Trevino—1st, 2nd, 3rd, 4th, 5th • Mr. Garza—1st, 3rd, 4th, 6th Four teachers offer Zane’s favorite computer class at different times during the day. The school counselor asks Zane if he prefers a morning or afternoon class. Below is a list of teachers and the periods they teach this class. The morning classes are 1st , 2nd, and 3rd periods, and the afternoon classes are 4th, 5th, and 6th periods. • Mr. Nelson—2nd, 4th, 5th, 6th • Ms. Trevino—1st, 2nd, 3rd, 4th, 5th • Mr. Garza—1st, 3rd, 4th, 6th • Ms. Jones—1st, 2nd, 3rd, 6th

1. Before answering the counselor’s question, Zane wants to list all the possibilities so he can make a choice that gives him the highest probability of getting a teacher he prefers. Create a table and a tree diagram that illustrate the possibilities. Create a table and a tree diagram that illustrate the possibilities. 1st period 2nd period 3rd period 4th period 5th period 6thperiod Mr. Nelson x x x x Ms. Trevino x x x x x Mr. Garza x x x x Ms. Jones x x x x

2. Before deciding on a morning or afternoon class, Zane remembers he wants to take his math class during 3rd period. What is the probability that he will be assigned the computer class during this time? Before considering morning or afternoon, there are 17 options for the computer class, 3 of which are 3rd period. The probability of getting assigned a computer class during 3rd period is 3/17 , or 17.6% 3. Zane prefers to be in the class of Ms. Trevino or Mr. Nelson. Should he pick the morning or the afternoon? P(Nelson in morning) =1/17 P(Nelson in afternoon) =3/17 P(Trevino in morning) =3/17 P(Trevino in afternoon) =2/17 P(Nelson in morning or Trevino in morning) =1/17 +3/17 =4/17 , or 23.5% P(Nelson in afternoon or Trevino in afternoon) =3/17 +2/17 =5/17 , or 29.4% Zane should pick the afternoon, since he has a slightly higher probability of getting Mr. Nelson or Ms. Trevino.

4. After checking the schedule, the counselor told Zane that Mr 4. After checking the schedule, the counselor told Zane that Mr. Garza’s classes are filled. How does this information affect the probability of Zane getting any afternoon class? If Zane asks for an afternoon class, how does this affect his probability of getting Mr. Nelson or Ms. Trevino? Overall, the probability of getting an afternoon class changes, since Mr. Garza’s classes are no longer an option. Now P(afternoon computer class) =6/13 , or 46.2%. The probability of getting Mr. Nelson or Ms. Trevino greatly improves because now there are fewer classes in the afternoon. The probability of getting Mr. Nelson or Ms. Trevino is now 5/6 , or 83.3%. 5. Because this is a required class for all students and Mr. Garza’s classes are filled, the school adds another teacher, Ms. Lopez. She will teach 1st and 6th periods. Does this fact affect the probability of getting Mr. Nelson in the morning? The probability of getting Mr. Nelson’s class changes. Zane now has a 1/8 or12.5% probability of getting Mr. Nelson if he asks for a morning class.

6. While Zane is calculating probabilities so he can make his decision, the class offerings change. (Mr. Garza’s classes fill, and Ms. Lopez is added.) If Zane requests an afternoon class, what is his probability of getting Ms. Trevino for 4th period? P(Trevino for 4th period) =1/7, or 14.3%