Probability of Rolling an Odd Prime Number on a Die

When a standard six-sided die is rolled, understanding the probability of rolling an odd prime number involves identifying the specific outcomes and applying basic principles of probability.

Sample Space and Outcomes

Consider a standard fair six-sided die. The sample space is {1, 2, 3, 4, 5, 6}. Among these, we identify the prime numbers, which are 2, 3, and 5. The odd prime numbers are those that are prime and odd, i.e., 3 and 5.

Single Die Rolling

When rolling a single die, the probability of getting an odd prime number (3 or 5) is given by:

Probability Number of favorable outcomes / Total number of possible outcomes

2 / 6

1 / 3

Multiple Dice Scenario

For a more complex scenario, consider rolling two fair six-sided dice. We need to determine the probability of getting at least one odd prime number.

Step-by-Step Calculation

Identify the odd numbers on a die: 1, 3, 5.

Identify the prime numbers among these odd numbers: 3, 5.

Determine the total outcomes when two dice are rolled: 6 x 6 36.

Calculate the favorable outcomes for at least one odd prime number:

No prime number: There are 4 non-prime numbers (1, 2, 4, 6) on each die. Therefore, there are 4 x 4 16 outcomes. At least one prime number: Total outcomes - outcomes with no prime numbers 36 - 16 20.

Thus, the probability of at least one odd prime number is:

Probability 20 / 36

5 / 9 ≈ 0.5556.

In the case of a single fair six-sided die, the probability of rolling an odd prime number is 1/3. When rolling two dice, the probability of getting at least one odd prime number is approximately 0.5556.

Further Proofs Using Set Theory

Let's use set theory to verify the probability calculation. Define:

A: Prime numbers on a die {2, 3, 5}

B: Odd numbers on a die {1, 3, 5}

Common elements: A ∩ B {3, 5}

Required probability P(A ∪ B) P(A) P(B) - P(A ∩ B)

3/6 3/6 - 2/6

4/6 2/3

This confirms that the probability of rolling an odd prime number on a single die is 2/3.

Conclusion

The probability of rolling an odd prime number on a six-sided die varies depending on whether one or two dice are rolled. For a single die, the probability is 1/3, while for two dice, it is approximately 0.5556. The prime and odd numbers on a standard six-sided die are 3 and 5, and there are no other numbers that satisfy both conditions.