Question

The phone lines to an airline reservation system are occupied 45% of the time. Assume that the events that the lines are occupied on successive calls are independent. Assume that eight calls are placed to the airline. What is the probability that for exactly two calls the lines are occupied? Round your answer to four decimal places

Answer 1

To find the probability that exactly two out of eight calls result in occupied lines with a 45% occupancy rate, use the binomial probability formula. After plugging in the values (n=8, k=2, p=0.45) and performing the calculations, the result is rounded to four decimal places.

Step-by-step explanation:

The question relates to the topic of probability in mathematics, specifically to the binomial probability distribution. The scenario describes phone lines being occupied 45% of the time and assumes independence between calls. With eight calls placed, we want to calculate the probability of exactly two calls resulting in occupied lines.

The binomial probability formula is used for this calculation:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

Where:

• P(X = k) is the probability of k successes in n trials
• C(n, k) is the number of combinations of n items taken k at a time
• p is the probability of success on a single trial
• n is the number of trials
• k is the number of successes (occupied lines in our case)

Here, p = 0.45, n = 8, and k = 2. Plugging in the values we get:

P(X = 2) = C(8, 2) * 0.45^2 * (1-0.45)^(8-2)

After performing the calculations, we round the result to four decimal places.

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