Probability Problem: Throwing a Fair Die
Data Science and A.I. Lecture Series
Problem Statement
A fair die is thrown. Find the probability of:
- A prime number
- An even number
- A number multiple of 2 or 3
- A number multiple of 2 and 3
- A number greater than 4
Step 1: Sample Space
Sample Space: The sample space \( S \) for throwing a fair die is:
\[
S = \{1, 2, 3, 4, 5, 6\}
\]
The total number of outcomes is:
\[
|S| = 6
\]
Step 2: Probability of a Prime Number
Prime Numbers: The prime numbers on a die are:
\[
\{2, 3, 5\}
\]
The number of favorable outcomes is:
\[
3
\]
The probability is:
\[
P(\text{Prime number}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2}
\]
Step 3: Probability of an Even Number
Even Numbers: The even numbers on a die are:
\[
\{2, 4, 6\}
\]
The number of favorable outcomes is:
\[
3
\]
The probability is:
\[
P(\text{Even number}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2}
\]
Step 4: Probability of a Number Multiple of 2 or 3
Multiples of 2 or 3: The numbers that are multiples of 2 or 3 are:
\[
\{2, 3, 4, 6\}
\]
The number of favorable outcomes is:
\[
4
\]
The probability is:
\[
P(\text{Multiple of 2 or 3}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{4}{6} = \frac{2}{3}
\]
Step 5: Probability of a Number Multiple of 2 and 3
Multiples of 2 and 3: The only number that is a multiple of both 2 and 3 is:
\[
\{6\}
\]
The number of favorable outcomes is:
\[
1
\]
The probability is:
\[
P(\text{Multiple of 2 and 3}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{6}
\]
Step 6: Probability of a Number Greater Than 4
Numbers Greater Than 4: The numbers greater than 4 on a die are:
\[
\{5, 6\}
\]
The number of favorable outcomes is:
\[
2
\]
The probability is:
\[
P(\text{Greater than 4}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{2}{6} = \frac{1}{3}
\]
Final Results
Summary:
- Probability of a prime number: \( \frac{1}{2} \)
- Probability of an even number: \( \frac{1}{2} \)
- Probability of a number multiple of 2 or 3: \( \frac{2}{3} \)
- Probability of a number multiple of 2 and 3: \( \frac{1}{6} \)
- Probability of a number greater than 4: \( \frac{1}{3} \)