Question

you have a product you are trying to sell and you are considering purchasing online ads. the cost of the ad will be \\$0.11 each time it is shown. the price of your product is \\$18.28. do the following for the given probability estimate.
a) find the expected value (to you) per ad that is viewed.
b) assume that you have an advertising budget large enough to place about 25,000 ads. are the ads likely to be a good purchase? explain.
c) would your answers to part (b) change if you could afford only 500 ads? explain.
based on your research, you believe that an average of 1 in 270 people who see your ad will buy your product.

b) assume that you have an advertising budget large enough to place about 25,000 ads. are the ads likely to be a good purchase? explain. choose the correct answer below.
a. yes, the ads will be a good purchase because the budget is already allocated for the ads.
b. no, they will not be a good purchase because the expected value is negative.
c. yes, they will be a good purchase due to the law of large numbers, and eventually they will pay off.
d. no, they will not be a good purchase because the price of the product is too high, not attracting enough customers.

c) would your answers to part (b) change if you could afford only 500 ads? explain. choose the correct answer below.
a. no, since the expected value is negative per ad, no matter the number of ads you can buy, this will be a poor purchase.
b. yes, this will be a good purchase because the expected value per ad when only 500 ads are purchased is positive.
c. no, since the law of large numbers only works with thousands of ads, this will be a poor choice.
d. yes, this will be a good purchase because the money is already allocated for the ads.

Explanation:

Calculate the expected value per ad

We first need to find the expected value per ad to understand the basis of the decisions.

• The cost of showing an ad is \(C = \$0.11\).
• The price of the product is \(P = \$18.28\).
• The probability of a purchase per ad view is \(p = \frac{1}{270}\).
• The probability of no purchase is \(1 - p = \frac{269}{270}\).

Using the Expected Value concept:

$$ E = p \times (P - C) + (1 - p) \times (-C) $$

$$ E = \frac{1}{270} \times (18.28 - 0.11) + \frac{269}{270} \times (-0.11) $$

$$ E = \frac{18.17}{270} - \frac{29.59}{270} = \frac{-11.42}{270} \approx -\$0.0423 $$

Since the expected value is negative, you expect to lose money on average for each ad shown.

Analyze part (b)

We evaluate the purchase of 25,000 ads.
Using the Law of Large Numbers, as the number of trials (ads shown) increases, the actual average outcome will converge to the expected value.
Since the expected value per ad is negative (\(\approx -\$0.04\)), buying a large number of ads (25,000) makes it highly likely that you will lose money overall.
Therefore, the ads are not a good purchase because the expected value is negative.
This matches option B for part (b).

Analyze part (c)

We evaluate if the decision changes if only 500 ads are purchased.
The expected value per ad remains negative (\(\approx -\$0.04\)) regardless of how many ads are purchased.
Even with a smaller sample size where fluctuations are more likely, the long-term expectation is still a loss, making it a poor investment.
Therefore, the answer does not change because the expected value is negative per ad.
This matches option A for part (c).

Answer:

Question 1

• A. Yes, the ads will be a good purchase because the budget is already allocated for the ads.
• B. No, they will not be a good purchase because the expected value is negative. (Correct answer)
• C. Yes, they will be a good purchase due to the law of large numbers, and eventually they will pay off.
• D. No, they will not be a good purchase because the price of the product is too high, not attracting enough customers.

Question 2

• A. No, since the expected value is negative per ad, no matter the number of ads you can buy, this will be a poor purchase. (Correct answer)
• B. Yes, this will be a good purchase because the expected value per ad when only 500 ads are purchased is positive.
• C. No, since the law of large numbers only works with thousands of ads, this will be a poor choice.
• D. Yes, this will be a good purchase because the money is already allocated for the ads.