Assume it is known that the probability of birth is equal in all months. What is the probability that in the STAT 1025 class of 120 students, exactly 20 students have their birthdays in either August or September? Solve using (i) the exact Binomial distribution, (ii) the Normal approximation to the Binomial distribution.
To calculate the probability using the exact Binomial distribution, we need to determine the probability of success and the number of trials.
In this case, the probability of success is the probability that a student's birthday falls in either August or September. Since it is known that the probability of birth is equal in all months, the probability of a student's birthday falling in either August or September is 2 out of 12, or 1/6. Thus, the probability of success is 1/6.
The number of trials is the total number of students in the STAT 1025 class, which is 120.
To calculate the probability using the exact Binomial distribution, we can use the following formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the number of trials
- k is the number of successes
- p is the probability of success
Substituting the given values into the formula, we have:
P(X = 20) = (120 choose 20) * (1/6)^20 * (5/6)^(120 - 20)
To calculate this value, you can use a calculator, statistical software, or an online binomial probability calculator.