Answer :
Final answer:
The expected maximum value of 10 balls picked from 20 is most reasonably the highest number possible, which is 20. This is because as we draw more balls without replacement, the probability of drawing the highest remaining number increases.So, d is the correct answer.
Explanation:
The question is asking to calculate the expected maximum value of the 10 balls picked randomly from a total of 20 balls labeled from 1 through 20. To tackle this probability question, we should first understand that the maximum value is always going to be the highest numbered ball picked, and since the balls are not replaced after being chosen, the probability of the next ball being the maximum increases as more balls are selected.
To find the expected value, we look at the probabilities of selecting each ball as the maximum. We start by assuming we picked 9 balls that are not the 20th, which can happen in C(19,9) ways. Then the 10th pick is the ball labeled '20' with a probability of 1/11 (since there are 11 balls left). The expected maximum value, therefore, tends to be higher as more balls are picked, because we exhaust the lower numbers first.
Without calculating an exact value, we can intuitively rule out answer choices a) 10 and b) 15, since these are within the range of our selection and would not typically be the maximum of 10 randomly picked balls from 1 to 20. Given the alternatives, c) 17 and d) 20, the tendency will be closer to 20 rather than an intermediate number because the higher numbers have a higher chance of appearing as the maximum in a random draw of 10 balls as we pick more from the urn. The exact expected value would require more detailed calculations, considering the varying probabilities of each ball being the highest, but for this scenario, the most logical choice would be the highest possible maximum, which is the ball withlabel20, leading us to conclude choice d) 20 is the expected maximum value.
