Find the probability that among five persons, at least two were born on the same weekday.
To find the probability that at least two persons out of five were born on the same weekday, we can use the concept of complementary probability.
Step 1: Find the total number of possible outcomes when assigning weekdays to the five persons. Since each person can be assigned any one of the seven weekdays, the total number of possible outcomes is 7^5 (7 to the power of 5).
Step 2: Find the number of favorable outcomes, i.e., the number of ways in which at least two persons are born on the same weekday.
To calculate this, we can use the principle of inclusion-exclusion. Let's consider the following cases:
- Case 1: Two persons are born on the same weekday: We have 7 ways to choose the weekday, and then we need to assign these weekdays to the two persons. The remaining three persons can be assigned any weekday, so there are (7 C 1) * (5 C 2) * (2 C 2) ways to do this.
- Case 2: Three persons are born on the same weekday: We have 7 ways to choose the weekday, and then we need to assign these weekdays to the three persons. The remaining two persons can be assigned any weekday, so there are (7 C 1) * (5 C 3) * (2 C 2) ways to do this.
- Case 3: Four persons are born on the same weekday: We have 7 ways to choose the weekday, and then we need to assign these weekdays to the four persons. The remaining person can be assigned any weekday, so there are (7 C 1) * (5 C 4) * (2 C 1) ways to do this.
- Case 4: All five persons are born on the same weekday: We have 7 ways to choose this weekday.
To calculate the number of favorable outcomes, we sum up the number of ways for each case: (7 C 1) * (5 C 2) * (2 C 2) + (7 C 1) * (5 C 3) * (2 C 2) + (7 C 1) * (5 C 4) * (2 C 1) + 7.
Step 3: Calculate the probability of at least two persons being born on the same weekday by dividing the number of favorable outcomes by the total number of possible outcomes.
Probability = Number of favorable outcomes / Total number of possible outcomes
Now, you can substitute the values we calculated in steps 1 and 2 to find the final probability.