What is the probability that 2 out of 8 students will have the same birthday?
P[ at least 2 people have birthday in same month]
1−P[no two people have birthday on same month] =1−[(12*11*10*9*8*7]/12^8 )= .99845
To calculate the probability that 2 out of 8 students will have the same birthday, we need to consider all the potential combinations of students' birthdays and determine how many of them have at least 2 students with the same birthday.
Step 1: Calculate the total number of possible combinations for the 8 students' birthdays. Since each student can have a birthday on any of the 365 days of the year (ignoring leap years), we have:
365^8 = 8,366,848,000,000,000 possible combinations.
Step 2: Calculate the number of combinations where all the students have different birthdays.
First student can have any of the 365 possible birthdays. The second student can have one of the remaining 364 possible birthdays, the third student has 363 options, and so on.
The number of combinations with all different birthdays is:
365 * 364 * 363 * ... * (365 - 7) = 365!/((365-8)!).
Step 3: Calculate the number of combinations where at least 2 students have the same birthday.
The total number of combinations with at least 2 students having the same birthday is the difference between the total number of combinations and the number of combinations where all students have different birthdays.
Number of combinations = Total combinations - Combinations with no matching birthdays.
Step 4: Calculate the probability.
The probability is the number of combinations with at least 2 students having the same birthday divided by the total number of combinations.
Probability = Number of combinations / Total combinations.
Let's calculate the probability.
To calculate the probability that 2 out of 8 students will have the same birthday, you can use the principle of complementary probability, which means finding the probability of the opposite event and subtracting it from 1.
Step 1: Calculate the probability that none of the students have the same birthday.
The first student can have any birthday, so the probability is 365/365 (assuming a non-leap year).
The second student must have a different birthday from the first, so the probability is 364/365.
Similarly, the third student's probability is 363/365, and so on, until the eighth student, whose probability is 358/365.
To find the probability of all students having different birthdays, we multiply these individual probabilities together:
(365/365) * (364/365) * (363/365) * (362/365) * (361/365) * (360/365) * (359/365) * (358/365) ≈ 0.6967
Step 2: Subtract the probability we just calculated from 1 to get the probability that 2 out of 8 students will have the same birthday:
1 - 0.6967 = 0.3033 or approximately 30.33%
Therefore, the probability that 2 out of 8 students will have the same birthday is approximately 30.33%.