Answer: 20
Explanation:
To determine the expected amount of money in the bag when it is claimed by someone, we can analyze the situation in terms of probabilities and expected values.
Let's break down the possible time intervals into 10-minute segments and calculate the probabilities for each segment.
1. In the first 10 minutes, the bag remains untouched with a probability of 1 (since no one has claimed it yet). The amount in the bag remains $10.
2. In the next 10 minutes (10 to 20 minutes), the bag has a 1/2 probability of remaining untouched, and a 1/2 probability of being claimed. If it remains untouched, the amount doubles to $20. If it is claimed, the person receives the current contents of $10.
3. In the following 10 minutes (20 to 30 minutes), the bag has a 1/4 probability of remaining untouched (since two consecutive 1/2 probabilities are multiplied together). If it remains untouched, the amount doubles again to $40. If it is claimed, the person receives the current contents of $20.
We can continue this analysis for subsequent 10-minute intervals:
4. 30 to 40 minutes: Probability of remaining untouched = 1/8, amount if untouched = $80.
5. 40 to 50 minutes: Probability of remaining untouched = 1/16, amount if untouched = $160.
6. 50 to 60 minutes: Probability of remaining untouched = 1/32, amount if untouched = $320.
We observe a pattern here: The probability of the bag remaining untouched in each 10-minute interval is halved compared to the previous interval. The amount in the bag if it remains untouched doubles in each interval.
To calculate the expected value, we can sum up the possible outcomes weighted by their probabilities:
Expected value = ($10 * 1) + ($20 * 1/2) + ($40 * 1/4) + ($80 * 1/8) + ($160 * 1/16) + ($320 * 1/32) + ...
This is an infinite geometric series with a common ratio of 1/2 (each term is half of the previous term). The formula for the sum of an infinite geometric series is:
Sum = a / (1 - r)
where 'a' is the first term and 'r' is the common ratio.
Plugging in the values, we get:
Expected value = ($10 * 1) / (1 - 1/2) = $10 / (1/2) = $20
Therefore, the expected amount of money in the bag when it is claimed by someone is $20.