the average proportion of red candies in a pakage is 20% and an 8 oz package contains about 250 candies.What is the probability that a randomly selected 8 oz bag contains less than 45 red candies?

To solve this problem, we need to use the binomial distribution, which is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success.

Let's define the following variables:

X: the number of red candies in an 8 oz package
n: the total number of candies in the package (250)
p: the probability of getting a red candy (0.2)
The probability of getting less than 45 red candies can be calculated as follows:

P(X < 45) = Σ(k=0 to 44) (n choose k) * p^k * (1-p)^(n-k)

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items.

Using a calculator or statistical software, we can calculate this probability to be approximately 0.00026, or 0.026%. This means that there is a very low probability of getting less than 45 red candies in an 8 oz package.

Therefore, if a package has less than 45 red candies, it may be considered to be defective or not meeting the expected standards. The manufacturer may need to investigate the cause of this issue and take corrective actions to improve the quality of their product.

To find the probability that a randomly selected 8 oz bag contains less than 45 red candies, we can use the concept of binomial distribution and the cumulative distribution function.

Let's break down the problem step by step:

Step 1: Determine the mean and standard deviation of the distribution.
Given that the average proportion of red candies is 20% and there are 250 candies in the bag, we can calculate the mean (µ) and standard deviation (σ) as follows:
µ = 250 * 0.20 = 50
σ = sqrt(250 * 0.20 * (1 - 0.20)) = sqrt(40) ≈ 6.32455

Step 2: Calculate the probability of having less than 45 red candies.
Now that we have the mean and standard deviation, we can use the cumulative distribution function (CDF) of the normal distribution to find the probability. However, since the binomial distribution can be approximated as a normal distribution when the sample size is large, we can use a continuity correction to approximate this binomial probability as a normal probability.
Using continuity correction, we adjust the score of 45 to the midpoint between 44.5 and 45.5, which is 45.

Step 3: Using the Z-score table to find the probability.
We need to calculate the Z-score, which represents the number of standard deviations away from the mean.
Z = (43.5 - 50) / 6.32455 ≈ -1.03
Using the Z-score table (or a calculator), we can find the corresponding probability associated with the Z-score of -1.03. The table (or calculator) will give us the value of 0.1522.

The calculated probability is approximately 0.1522, which represents the probability that a randomly selected 8 oz bag contains less than 45 red candies.