Probability

Let: = Number of truth-tellers selected when 2 people are chosen at random from a group.
Suppose there are 50 people: 20 truth-tellers and 30 liars.
Possible values of : 0, 1, 2
Case 1: (Both are liars)
Case 2: (Both are truth-tellers)
Case 3: (One truth-teller, one liar)
Two ways: Truth then lie OR Lie then truth
Therefore, the probability distribution is:

Let the students be .

Case 1: None solves the problem

Case 2: Exactly one solves the problem

Case 2.1: Only solves

Case 2.2: Only solves

Case 2.3: Only solves

Case 2.4: Only solves

Total for exactly one solves:

Final Answer:

Expected Value and Probability from Distribution

Given:

Expected value

Step 1: Write the expression for

Step 2: Simplify constant and variable terms

Constants:

Convert to denominator 50:

Variable terms:

Full expression for expected value:

Step 3: Use given value

Equating:

Step 4: Find

Values of are: 1, 2, 4

✅ Final Answers:

Total outcomes when a die is thrown 3 times: Let be the event: number occurs at least once. Numbers > 4 are: 5 and 6 favorable outcomes = 2 So, let’s find complement event: no number > 4 That means all outcomes are from {1,2,3,4} So each roll has 4 options: Final Answer: