For each of the last 100 days at a certain factory, Light A was either on or off for that...

Phase 1: Owning the Dataset

We have 100 days with two lights (A and B) that can each be on or off. Let's create a table to organize the possible states:

Now let's apply our constraints:

• Constraint 1: If A is off → B is off
• Constraint 2: If B is on → A is on

These constraints tell us that B can ONLY be on when A is also on. This means "Only B on" is impossible!

Phase 2: Understanding the Question

Given information:

• For exactly 40 of 100 days: at least one light was on
• For exactly 20 of 100 days: both lights were on

Let's update our table with what we can deduce:

Key insight: Since B can only be on when A is on:

• Days with "at least one light on" = Days with "A on" = 40
• These 40 days split into: 20 days (only A) + 20 days (both A and B)

Phase 3: Finding the Answer

Now we can calculate the probabilities:

For \(\mathrm{P(A)}\): Light A was on for:

• 20 days (only A on) + 20 days (both on) = 40 days
• \(\mathrm{P(A)} = \frac{40}{100} = 0.40\)

For \(\mathrm{P(B)}\): Light B was on for:

• 0 days (only B on) + 20 days (both on) = 20 days
• \(\mathrm{P(B)} = \frac{20}{100} = 0.20\)

Phase 4: Solution

Final Answer:

• \(\mathrm{P(A)} = 0.40\)
• \(\mathrm{P(B)} = 0.20\)

These values satisfy all our constraints and match the given information perfectly.