Area Models and Probability – Page 316
Andrew has 2 choices when taking the bus:
- Route 3 – passes by the bus stop every 20 minutes
- Route 5 – passes by the bus stop every 15 minutes
He will get soaked if he waits for more than 5 minutes. Find the probability that Andrew does NOT get soaked.
Experimental Probability
· Find the experimental probability by setting up a simulation with graphing calculators.
· Find a partner, one person be Route 3 bus and the other be Route 5 bus.
· Route 3 bus do RandInt(1,20) which will provide a random number between 1 and 20
· Route 5 bus do RandInt(1,15) which will provide a random number between 1 and 15
· If Andrew does NOT want to wait more than 5 minutes, what are we looking for in our numbers?
· Do 50 trials.
· What is your experimental probability of Andrew not getting soaked?? ______
Theoretical Probability
Use an area model to represent the theoretical probability of not getting soaked.
Andrew could spend from 1 to 20 minutes waiting for the Route 3 bus (x-axis).
Andrew could spend from 0 to 15 minutes waiting for the Route 5 bus (y-axis).
Complete the Venn Diagram. How are Venn diagrams and area models the same? Different?
The grid above represents all of the possible times that could
be spent waiting for the bus.
Find the following probabilities:
· P(route 3) = This is the probability of being picked up by the route 3 bus within 5 min.
Fraction that the 5 min region is of the total shaded region for route 3.
· P(route 5) = This is the probability of being picked up by the route 5 bus within 5 min.
Fraction that the 5 min region is of the total shaded region for route 5.
· P(route 3 and route 5) = This is the probability of both buses arriving within 5 minutes.
Fraction that the combined 5 minute region is of the total shaded region for both buses.
**P(route 3 and route 5) is the probability of both buses arriving within 5 minutes (not getting soaked and having two choices of bus)**
Theoretically, is this a mutually exclusive or mutually inclusive event? Explain.
Now find:
· P(route 3 or route 5) = What fraction of the TOTAL area is the SHADED region?
Compare this to P(route 3 or route 5)
Area Diagrams are a visual representation where the area used is proportional to the probabilities.
Investigation #7 – Alyssa and Ja-Wen’s Meeting Problem
Let’s use an area model to find the theoretical probability of these two girls meeting before entering the DMV office.
Details:
· both will arrive between 10:00 and 10:30
· neither will wait for more than 10 minutes
*If Alyssa arrives at 10:00, what is the latest time that Ja-Wen could arrive so that they would meet?
Ordered pair to represent this would be (0,____).
Plot this point.
*If Alyssa arrives at 10:05, what is the latest time that Ja-Wen could arrive so that they would meet?
Ordered pair to repesent this would be (5, ____).
Plot this point.
*Repeat this procedure for Alyssa and then do it for Ja-Wen and you will have graphed the region that represents the girls meeting!
What is the AREA of the shaded region? (hint: You could count squares but is there an easier way by considering the non-shaded region?)
What is the theoretical probability that the two friends will meet?
Page 318
#41. a) Use RED to shade the part of the grid that corresponds to rolling an odd on the red die.
b) Use BLUE to shade the part of the grid that corresponds to rolling an even on the blue die.
c) Use the area diagram to find:
i) P(odd and even) ii) P(odd or even)
#45. a) Use an area model to find the probability of rolling a number less than four on the green die and a total on both dice that is an even number.
b) Use the same area model to find probability of one OR the other occurring.