In the old days, we could buy 8 inch floppies for our computers on which to store our favorite programs and data. But, the manufacturing ability was not that good back then and 1% of these 8 inch floppies sold had a bad sector or two on them which made them have less memory space. If your company bought 1000 8 inch floppies for data storage and backup, what is the chance that none of them have a bad sector? Use the normal approximation to the binomial to calculate this probability. Remember, all probabilities are taken to 4 decimal places for this class
To calculate the probability that none of the 1000 8 inch floppies have a bad sector, we can use the binomial distribution and then approximate it using the normal distribution. Here's how you can do it step by step:
Step 1: Define the variables
Let's define the probability of getting a bad sector on a single floppy as p. Since 1% of the 8 inch floppies have a bad sector, p = 0.01. The number of floppies purchased, n, is 1000.
Step 2: Calculate the mean and standard deviation
The mean (μ) of the binomial distribution is μ = n * p = 1000 * 0.01 = 10. The standard deviation (σ) is given by σ = sqrt(n * p * (1 - p)) = sqrt(1000 * 0.01 * (1 - 0.01)) = sqrt(9.9) ≈ 3.146.
Step 3: Apply the normal approximation
To use the normal approximation, we need to convert the probability of none of the floppies having a bad sector to a z-score using the standard normal distribution table.
P(X = 0), where X follows a binomial distribution, can be approximated using the normal distribution as P(X < 0.5).
To convert it to a z-score, we calculate (0.5 - μ) / σ. In this case, (0.5 - 10) / 3.146 ≈ -3.17.
Using the standard normal distribution table, we can find the probability associated with this z-score. The table gives the area to the left of the z-score, so we will look up the absolute value of -3.17 and subtract that value from 0.5 to get the final probability.
Looking up the absolute value of -3.17 in the table, we find the area to the left is approximately 0.00075.
Finally, subtracting 0.00075 from 0.5, we get the probability that none of the 1000 8 inch floppies have a bad sector, which is approximately 0.49925.
Therefore, the chance that none of the 1000 8 inch floppies have a bad sector is approximately 0.4992 when rounded to 4 decimal places.