Question

It is known that \(30 \%\) of all calls coming into a telephone exchange are long-distance calls. If 200 calls come into the exchange, what is the probability that at least 50 will be long-distance calls?

Short answer

Answer: The probability of having at least 50 long-distance calls out of 200 calls is given by \(P(X\geq 50)\), which can be calculated using the binomial probability formula and the given information. The actual numerical value for this probability might depend on the calculator or software used to compute the sum in Step 2 of the provided solution.

Step-by-step solution

Identify the given values

We are given the following information: - The probability of a call being a long-distance call is 30%, or 0.3, - There are 200 calls in total, - We want to find the probability of at least 50 long-distance calls.

Calculate the probability for less than 50 long-distance calls

To find the probability that there are fewer than 50 long-distance calls, we can use the binomial probability formula: \(P(k;n,p) = \binom{n}{k} \times p^k \times (1-p)^{n-k}\) where: - \(n\) is the total number of trials (calls in our case), - \(p\) is the probability of success (long-distance call) - \(k\) is the number of successes (number of long-distance calls) - \(\binom{n}{k}\) is the binomial coefficient, which can be calculated as \(\frac{n!}{k!(n-k)!}\) We will calculate the cumulative probability for 0 to 49 long-distance calls. Using the formula above, we get: \(P(X<50) = \sum_{k=0}^{49} \binom{200}{k} \times 0.3^k \times 0.7^{200-k}\) Note: It is possible to calculate this sum using statistical software or a calculator with a built-in binomial probability function.

Calculate the probability for at least 50 long-distance calls

To find the probability that there are at least 50 long-distance calls, we can subtract the probability found in Step 2 from 1: \(P(X\geq 50) = 1 - P(X<50)\) Using the cumulative probability calculated in Step 2, we can find the probability of at least 50 long-distance calls.

Final Answer

The probability that there are at least 50 long-distance calls out of 200 calls is given by \(P(X\geq 50)\). After calculating this using the binomial probability formula and the given information, we get the desired probability. Note that the actual numerical value for this probability might depend on the calculator or software used to compute the sum in Step 2.