In Exercises 21-28, find the probability and answer the questions.


Guessing Birthdays On their first date, Kelly asks Mike to guess the date of her birth, not including the year.


b. Would it be unlikely for him to guess correctly on his first try?

Verified step by step guidance

1. Step 1: Understand the problem. Kelly is asking Mike to guess her birth date (month and day) without including the year. There are 12 months in a year, and each month has a specific number of days. To calculate the probability, we need to determine the total number of possible dates in a year.
2. Step 2: Calculate the total number of possible dates. Assuming a non-leap year, there are 365 days in a year. Each day represents a unique combination of month and day. Therefore, the total number of possible dates is 365.
3. Step 3: Determine the probability of guessing correctly on the first try. Probability is calculated as the ratio of favorable outcomes to total outcomes. In this case, there is only 1 favorable outcome (guessing Kelly's exact birth date) out of 365 possible outcomes. The probability formula is: $\frac{1}{365}$.
4. Step 4: Interpret the probability value. A probability of $\frac{1}{365}$ is very small, indicating that it is highly unlikely for Mike to guess correctly on his first try.
5. Step 5: Conclude the answer. Based on the calculated probability, it would indeed be unlikely for Mike to guess Kelly's birth date correctly on his first try.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the context of guessing a birthday, the probability of correctly guessing a specific date can be calculated by considering the total number of possible outcomes, which is typically 365 for a non-leap year. Thus, the probability of guessing correctly on the first try is 1/365, indicating a very low likelihood.

Sample Space

The sample space is the set of all possible outcomes of a random experiment. In this scenario, the sample space consists of all the days in a year, which is 365 days. Understanding the sample space is crucial for calculating probabilities, as it provides the basis for determining how many favorable outcomes exist compared to the total number of outcomes.

Unlikely Events

An event is considered unlikely if its probability is significantly low, often defined as less than 0.05 (or 5%). In the case of Mike guessing Kelly's birthday, with a probability of 1/365, this event falls well below the 5% threshold, making it unlikely for him to guess correctly on his first try. Recognizing unlikely events helps in understanding the nature of chance and randomness in probability.