Question

As of 2021, 35% of all vehicles on the road were white. Suppose a random sample is taken of 300 cars. What is the probability that at most 30% are white? Must show all work.

Answer 1

To find the probability that at most 30% of the 300 cars in a random sample are white, we can use the binomial distribution and calculate the probability using the binomial probability formula.

Step-by-step explanation:

To find the probability that at most 30% of the 300 cars in a random sample are white, we need to use the binomial distribution. Let's define a success as a car being white and a failure as a car being any color other than white. The probability of success is 35% (0.35) and the probability of failure is 65% (1 - 0.35). We can use the binomial probability formula to calculate the probability:

P(X ≤ k) = ∑ (k = 0 to k)

Where:

• P(X ≤ k) is the probability that at most k successes occur
• n is the number of trials (300)
• p is the probability of success (0.35)
• k is the number of successes

Let's calculate the probability:

1. P(X ≤ 30) = ∑ (k = 0 to 30) [300Ck * (0.35)^k * (1 - 0.35)^(300 - k)]
2. Using a binomial distribution calculator or software, we find that P(X ≤ 30) ≈ 0.6538

Therefore, the probability that at most 30% of the cars in the random sample are white is approximately 0.6538 or 65.38%.