In 2011, Six Flags St. Louis had 10 roller coasters: The...

In 2011, Six Flags St. Louis had 10 roller coasters: The Screamin' Eagle, The Boss, River King Mine Train, Batman the Ride, Mr. Freeze, Ninja, Tony Hawk's Big Spin, Evel Knievel, Xcalibur, and Sky Screamer. Of these, The Boss, The Screamin' Eagle, and Evel Knievel are wooden coasters. Ethan wants to ride two more roller coasters before leaving the park (not the same one twice) and decides to select them by drawing names from a hat. (a) Determine the sample space of the experiment. That is, list all possible simple random samples of size $n=2$. (b) What is the probability that Ethan will ride $\mathrm{Mr}$. Freeze and Evel Knievel? (c) What is the probability that Ethan will ride the Screamin' Eagle? (d) What is the probability that Ethan will ride two wooden roller coasters? (e) What is the probability that Ethan will not ride any wooden roller coasters?

In 2011, Six Flags St. Louis had 10 roller coasters:
The Screamin' Eagle, The Boss, River King Mine Train, Batman the Ride, Mr. Freeze, Ninja, Tony Hawk's Big Spin, Evel Knievel, Xcalibur, and Sky Screamer. Of these, The Boss, The Screamin' Eagle, and Evel Knievel are wooden coasters. Ethan wants to ride two more roller coasters before leaving the park (not the same one twice) and decides to select them by drawing names from a hat.
(a) Determine the sample space of the experiment. That is, list all possible simple random samples of size $n=2$.
(b) What is the probability that Ethan will ride $\mathrm{Mr}$. Freeze and Evel Knievel?
(c) What is the probability that Ethan will ride the Screamin' Eagle?
(d) What is the probability that Ethan will ride two wooden roller coasters?
(e) What is the probability that Ethan will not ride any wooden roller coasters?

Key Concepts

Probability Calculation

Probability calculation is the process of determining how likely an event is, by taking the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. This concept is used to compute the likelihood of various events such as selecting specific roller coasters or pairs of wooden roller coasters, by identifying the favorable cases and comparing them to the overall sample space.

Combinatorial Analysis

This concept involves counting the number of ways to choose items from a set where the order does not matter and repetitions are not allowed, typically using combinations (n choose k). It provides the mathematical foundation for determining the total number of possible pairs of roller coasters, as well as the number of favorable pairs for any event in the experiment.

Sample Space

The sample space is defined as the set of all possible outcomes of an experiment. In this context, it consists of every unique pair of two different roller coasters that can be selected from the park. Establishing the sample space is a crucial step when calculating probabilities, as it sets the basis for comparing favorable outcomes against all outcomes.

Transcript

00:03 In 2011, six flags st.

00:05 Louis had 10 roller coasters, the screaming eagle, the boss, the river king, mind train, batman, the ride, mr.

00:13 Freeze, ninja, tony hawk's big spin, and evil coneval.

00:19 Of these, the boss, the screaming eagle, and evil coneval are wooden coasters.

00:24 Ethan wants to ride two more roller coasters before leaving the park, not the same one twice, and decides to select them by drawing names from a hat.

00:32 Determine the sample space of the experiment.

00:34 Now this is a large sample space, so i'm going to show this more visually than trying to list them out.

00:44 So there are 10 coasters.

00:59 All right, so let's pretend that one is the screaming eagle.

01:06 Two is the boss.

01:10 Then it's river king, mind train, batman, the ride, mr.

01:17 Freeze, ninja, tony hawk's big spin, evil caneval, excal.

01:26 And skyscreamer.

01:29 And in this representation here, these would be the same.

01:34 Now, we don't want to ride the same coaster twice, so i'm going to put an x in this diagonal.

01:44 So 4 -4, 55, 6, 6, 7, 7, all the way down.

01:54 But every place else would be an example of the combinations.