Questions: Determine the probability that a randomly selected person does not have a birthday of the 31^text st day of a month.

Determine the probability that a randomly selected person does not have a birthday of the 31^text st day of a month.
Transcript text: Determine the probability that a randomly selected person does not have a birthday of the $31^{\text {st }}$ day of a month.
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Solution

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Solution Steps

To determine the probability that a randomly selected person does not have a birthday on the 31st day of a month, we first need to calculate the total number of days in a year and the number of days that are the 31st. Then, we find the probability of a birthday being on the 31st and subtract it from 1 to get the probability of not having a birthday on the 31st.

Step 1: Determine the Total Number of Days in a Year

In a non-leap year, there are a total of 365 days.

Step 2: Count the Number of 31st Days in a Year

There are 7 months with 31 days: January, March, May, July, August, October, and December. Therefore, there are 7 days in a year that are the 31st.

Step 3: Calculate the Probability of Having a Birthday on the 31st

The probability that a randomly selected person has a birthday on the 31st day of a month is given by the ratio of the number of 31st days to the total number of days in a year: \[ P(\text{31st}) = \frac{7}{365} \approx 0.01918 \]

Step 4: Calculate the Probability of Not Having a Birthday on the 31st

The probability that a randomly selected person does not have a birthday on the 31st day of a month is the complement of the probability of having a birthday on the 31st: \[ P(\text{not 31st}) = 1 - P(\text{31st}) = 1 - 0.01918 \approx 0.9808 \]

Final Answer

\[ \boxed{\frac{358}{365}} \]

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