Tennis Replay In a recent year, there were 879 challenges made...

Question

Tennis Replay In a recent year, there were 879 challenges made to referee calls in professional tennis singles play. Among those challenges, 231 challenges were upheld with the call overturned. Assume that in general, 25% of the challenges are successfully upheld with the call overturned.

a. If the 25% rate is correct, find the probability that among the 879 challenges, the number of overturned calls is exactly 231.

a. If the 25% rate is correct, find the probability that among the 879 challenges, the number of overturned calls is exactly 231.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says during a particular month, a call center received 500 calls about a specific issue. Assuming that historically, 40% of such calls are resolved on the first attempt, find the probability that exactly 210 calls were resolved on the first attempt, and we're given four possible choices as our answers. For choice A, we have 0.0123, For choice B, we have 0.0357. For Choice C, we have 0.0237, and for choice D, we have 0.0645. Now, we're asked to calculate the probability that exactly 210 calls were resolved on the first attempt. So, in order to calculate the probability, we're first going to calculate the Z scores for the costs around 210, and then use those to calculate the probability. Now, in order to calculate the Z score, we'll need to have our mean and standard deviation. So the first thing we'll do is calculate our mean, which is going to be mute, and recall that that's going to be equal to N multiplied by P. Where in here is the total number of calls, and P is the probability of resolving such calls on the first attempt. And so this is going to be equal to 500, since that is the total number of calls that we received, multiplied by the probability of resolving the call on the first attempt. And we're told that that is 40%, and so we'll need that as a decimal for a probability, so dividing that percentage by 100%, we'll get 0.4%. And so that means that our average mu here is going to be equal to. 200 Next, we'll need to calculate our standard deviation sigma. And that's going to be equal to, and here recall your formula for the standard deviation, that's going to be the square root of quantity of N multiplied by P, multiplied by the quantity of 1 minus P in quantity. And so this is going to be equal to the square root of the quantity of 500. Multiplied by 0.4. Multiplied by the quantity of 1 minus 0.4 in quantity. And when we evaluate the expression, we will get 10.954. So here we just rounded to three decimal places. So now we need to use this information to calculate our Z score. So we call that the Z score. Z is going to be equal to the quantity of X. Minus U in quantity divided by sigma. So we're actually going to have to calculate two Z scores, one that corresponds with X equal to 209.5, and another one where X is equal to 210.5. And the reason why we need two of those Z scores is we'll have the Z scores for those two values, and then actually use those Z scores to find the probabilities for those Z scores and then subtract those two probabilities to get the probabilities of finding exactly 210 calls. So, the first one we'll need to calculate is for Z of 209.5. And this is going to be equal to the quantity of 209.5 minus 200. And that quantity is going to be divided by our standard deviation of 10.954. And when we evaluate this expression, this is going to be equal to. 0.87. The next Z value that we'll calculate is going to be Z when X is equal to 210.5, and this is going to be equal to the quantity of 210.5 minus 200 in quantity, divided by. 10.954 And by this expression, this is going to be equal to. 0.96. Now we need the probabilities for the Z values. So the first one we'll look at is the probability. Of our the value for 209.5. And that's going to be equal to the probability of 0.87. Which we use our distribution tables, this will give us. 0.8078. Next, we'll need to find our probability. With a devalue that we found for the 210.5 case. And so this is going to be equal to probability of 0.96, which again, using our tables, this is going to be equal to. 0.8315. So now we need to use these values to calculate the probability of finding exactly 210 calls. So, the probability of finding exactly 210 calls is going to be equal to the Probability Of having a Z value with a number of calls of 210.5. Minus the probability of finding the Z value when we had 200. It's 209.5 calls. And the reason why we chose those two, because they're very close together and it's very close to 210. And so by subtracting these two, we will get the probability of getting exactly 210 calls. And so this is going to be equal to 0.8315 minus 0.8078. And evaluating this expression, this is going to be equal to. 0.0237. So this here is the answer that we are looking for for this problem. And if we compare our answer here to our answer choices, the correct answer choice corresponds to answer choice C. So, I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Binomial Distribution

Probability Mass Function (PMF)

Normal Approximation to the Binomial

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