Understanding the Problem
To calculate the probability that 4 randomly selected children have different birthdays, we assume there are 365 days in a year (ignoring leap years).
Calculating Total Outcomes
- Each child can be born on any of the 365 days.
- Therefore, the total possible outcomes for 4 children is:
- \( 365 \times 365 \times 365 \times 365 = 365^4 \)
Calculating Favorable Outcomes
- For the first child, there are 365 options.
- For the second child, to ensure a different birthday, there are 364 options.
- For the third child, there are 363 options.
- For the fourth child, there are 362 options.
- Thus, the number of favorable outcomes where all children have different birthdays is:
- \( 365 \times 364 \times 363 \times 362 \)
Calculating the Probability
- The probability \( P \) that all 4 children have different birthdays can be calculated as:
\[
P = \frac{365 \times 364 \times 363 \times 362}{365^4}
\]
- Simplifying this gives:
\[
P = \frac{364}{365} \times \frac{363}{365} \times \frac{362}{365}
\]
- This represents the fraction of favorable outcomes to total outcomes.
Final Result
- By calculating the above expression, you will find the probability that all 4 children have different birthdays, which is approximately 0.9836, or 98.36%.
This indicates that there is a high likelihood that 4 randomly selected children will have different birthdays.