📸 Example 2.2 - Camera Specifications – Understanding the Sample Space

📐 Case 1: Measuring Exact Times – ℝ⁺ × ℝ⁺

If you're recording the actual recycle time of two cameras, the outcomes are real numbers greater than zero.

Mathematically, the sample space is written as: S = ℝ⁺ × ℝ⁺

🧪 Click to Learn What ℝ⁺ × ℝ⁺ Means

ℝ⁺ means "positive real numbers" like 1.2, 3.5, 9.99.

So ℝ⁺ × ℝ⁺ means a pair like (2.1 sec, 1.5 sec) where each number is the recycle time for one camera.

This is a continuous sample space because there are infinitely many combinations.

✅ Case 2: Did the Cameras Pass or Fail?

We don’t care about exact numbers anymore. Just whether each camera passed the test.

Use "y" for yes (passed) and "n" for no (failed). Then the sample space is:

S = {yy, yn, ny, nn}

📘 Click to See What Each Means

This is a discrete sample space with only 4 outcomes.

🔢 Case 3: Count Cameras That Passed

If we only care about how many cameras passed, the outcomes are:

S = {0, 1, 2}

This is a simplified discrete sample space.

♾️ Case 4: Keep Testing Until One Fails

We test cameras one by one until we find one that fails.

Sample space: S = {n, yn, yyn, yyyn, ...}

🔍 What Does This Mean?

This is called a countably infinite discrete sample space.

📷 Visual Summary: Sample Space Types

1. ℝ⁺ × ℝ⁺ – Continuous Sample Space

This is a graph where both axes represent positive real numbers (like 1.5, 3.2, etc.). Imagine a shaded area in the top-right quadrant — each point (x, y) is a valid outcome like (2.1 sec, 3.4 sec).

Type: Continuous

Use Case: Measuring exact times or sensor readings

2. Discrete Sample Space – Pass/Fail

Only 4 outcomes exist: yy, yn, ny, nn.

Type: Discrete

Use Case: Test result combinations

3. Countably Infinite – Keep Testing Until Fail

We test one by one until failure: n, yn, yyn, yyyn, ...

This is an endless list, but you can count them one by one. Example:

Type: Countably infinite discrete

Use Case: Reliability or stress testing

📝 Student Activity: Apply What You Learned

Choose one of the following real-world situations and explain how the concept of sample space applies. Use terms like outcome, discrete vs. continuous, or countably infinite sample space.

  1. Online Exam + Power Outage: You are taking an online statistics exam, and the power goes out unexpectedly. How would you model the outcomes?
  2. Machine Testing Bolts: A machine checks bolts for strength and records pass/fail for each one. What does the sample space look like if we stop when we find the first failed bolt?
  3. Bus Arrival Times: You're recording the arrival time (in minutes) of a campus shuttle over 3 days. How is this sample space different from the bolt or camera examples?

Answer Key — Apply What You Learned

Online Exam + Power Outage. The sample space can be modeled as S = {lost power, did not lose power}, which is discrete. An outcome is whichever of the two happens for a chosen person. The event “lost power” is the subset {lost power}. If 3 of 80 people lost power, an empirical probability is P(lost power) = 3/80 ≈ 0.0375. If you select a person at random, that selection is a random experiment with outcomes in S.

Machine Testing Bolts. If you stop when the first failure occurs, the sample space is countably infinite and discrete: S = {n, yn, yyn, yyyn, …}, where y = pass and n = fail. Each outcome is a finite string of passes ending in a fail. An event might be “first failure after k passes,” i.e., {y…yn}. With constant pass probability per bolt, this aligns with a geometric-type model.

Bus Arrival Times. Over 3 days, one way to model is a continuous sample space of real-valued arrival times, e.g., S ⊆ ℝ × ℝ × ℝ (restricted to the service window each day). An outcome is a triple like (7.2, 9.0, 6.8) minutes. An event could be “arrives within 5 minutes on all 3 days,” which corresponds to a rectangular region in that 3-D space. This differs from the pass/fail examples because any real time in the interval is possible.