Answer: The problem involves a binomial distribution with n = 100 candies and p = 0.2 probability of success (getting an orange candy).
To find the probability that there are 25% or less orange candies, we need to calculate the cumulative probability of getting 0 to 25 orange candies out of 100:
P(X ≤ 25) = Σ P(X = k), where k goes from 0 to 25
Using the binomial probability formula, we get:
P(X = k) = (100 choose k) * 0.2^k * 0.8^(100-k)
So, the probability of getting 25% or less orange candies is:
P(X ≤ 25) = Σ (100 choose k) * 0.2^k * 0.8^(100-k), where k goes from 0 to 25
Using a calculator or software, we can compute this probability to be approximately 0.058.
To find the probability that there are 25% or more orange candies, we need to calculate the cumulative probability of getting 25 to 100 orange candies out of 100:
P(X ≥ 25) = Σ P(X = k), where k goes from 25 to 100
Using the same binomial probability formula, we get:
P(X = k) = (100 choose k) * 0.2^k * 0.8^(100-k)
So, the probability of getting 25% or more orange candies is:
P(X ≥ 25) = Σ (100 choose k) * 0.2^k * 0.8^(100-k), where k goes from 25 to 100
Using a calculator or software, we can compute this probability to be approximately 0.982.
Therefore, the probability of getting 25% or less orange candies is 0.058 and the probability of getting 25% or more orange candies is 0.982
Explanation: