Question

use the theoretical method to determine the probability of the following event.
sharing a birthday with another person when you both have birthdays in january.
(type an integer or a simplified fraction.)

Explanation:

Identify the sample space and outcomes

Using the Theoretical Probability knowledge point, we first determine the total number of possible outcomes for the two birthdays. January has 31 days. Since each person can have a birthday on any of the 31 days, the total number of possible birthday pairs for two people is:

$$ 31 \times 31 = 961 $$

Determine the favorable outcomes

For the two people to share a birthday, they must both be born on the exact same day of January. There are 31 possible days in January on which this shared birthday can occur (January 1st, January 2nd, ..., January 31st). Thus, there are 31 favorable outcomes.

Calculate the theoretical probability

Using the Theoretical Probability formula, we divide the number of favorable outcomes by the total number of possible outcomes:

$$ P(\text{sharing a birthday}) = \frac{31}{961} $$

Simplify the fraction

We simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 31:

$$ \frac{31 \div 31}{961 \div 31} = \frac{1}{31} $$

Answer:

Use the theoretical method to determine the probability of the following event.
Sharing a birthday with another person when you both have birthdays in January.
<blank>\(\frac{1}{31}\)</blank> (Type an integer or a simplified fraction.)