Suppose that a sample of n = 1,600 tires of the same type are obtained at random from an ongoing production process in which 8% of all such tires produced are defective. What is the probability that in such a sample 150 or fewer tires will be defective
thats hard
To find the probability that 150 or fewer tires will be defective, we will use the binomial distribution formula.
The binomial distribution formula is given by:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes (in this case, defective tires)
- (n C k) represents the number of combinations of n items taken k at a time
- p is the probability of a single success (probability of a defective tire)
- q = (1 - p), which is the probability of a single failure (probability of a non-defective tire)
- n is the total number of trials (number of tires in the sample)
- k is the number of successful trials (number of defective tires)
Given: n = 1,600, p = 0.08, and we need to calculate the probability of getting 150 or fewer defective tires (k ≤ 150).
To solve this problem, we need to find the cumulative probability P(X ≤ 150) using the binomial distribution formula for each value of k from 0 to 150, and then sum up all of these individual probabilities.
P(X ≤ 150) = P(X = 0) + P(X = 1) + ... + P(X = 150)
Calculating each individual probability may be time-consuming. However, we can use statistical software or an online calculator to calculate the cumulative probability directly.
For example, using an online binomial distribution calculator, we can enter the parameters n = 1600, p = 0.08, and k = 150, and it will give us the probability P(X ≤ 150) as the result.
The result is approximately 0.9999999999999994, or in decimal form, 1.0000 (rounded to four decimal places).
Therefore, the probability that 150 or fewer tires will be defective is 1.0000 or 100%.