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 solve this problem, we'll calculate the probability using both the exact Binomial distribution and the Normal approximation to the Binomial distribution.
(i) Exact Binomial distribution:
The probability of exactly 20 students having their birthdays in either August or September can be calculated using the Binomial distribution formula.
The formula for the Binomial distribution is given by:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of exactly k successes (in this case, 20 students with birthdays in August or September)
- n is the number of trials (students in the class, 120)
- k is the number of successes (students with birthdays in August or September, 20)
- C(n, k) is the binomial coefficient, calculated as C(n, k) = n! / (k! * (n-k)!)
- p is the probability of success (probability of birth in either August or September)
Given that the probability of birth is equal in all months, the probability of birth in either August or September is 2/12 = 1/6.
Using this information, we can calculate the probability using the formula:
P(X=20) = C(120, 20) * (1/6)^20 * (5/6)^(120-20)
(ii) Normal approximation to the Binomial distribution:
For large values of n (number of trials) and when both np and n(1-p) are greater than 5, we can use the Normal approximation to the Binomial distribution.
The mean (μ) of the Binomial distribution is given by μ = n * p, and the standard deviation (σ) is given by σ = sqrt(n * p * (1-p)).
Using these values, we can approximate the probability P(X=20) by finding the z-score and referring to the Z-table.
The z-score formula is:
z = (X - μ) / σ
Once we have the z-score, we can find the corresponding probability using the Z-table, or by using a calculator.
Now, let's calculate the probability using both approaches.