An observer arrives at time slot 10 and sees that no arrival took place in that slot. A passerby informs the observer that there was exactly one arrival during the preceding two time slots (i.e., time slots 8 and 9) but has no additional information about which of the two it was.
Let T be the number of time slots starting from the time slot of the last arrival (before slot 10) and until the time slot where the next arrival comes.
For example, if the last arrival came at time 8 and the next arrival comes at time 13, T is the number of slots in the interval {8,9,10,11,12,13}, so that T=6.
(Enter a numerical expression or a decimal accurate to at least 3 decimals places.)
Find the probability that T=4.
unanswered
Problem 4(b)
2.0 points possible (graded, results hidden)
Continue from and use the same setup as above.
Find the expected value of T.
To find the probability that T=4, we need to consider the possible scenarios for the arrival in the preceding two time slots.
There are two possible scenarios:
1. Arrival in time slot 8, no arrival in time slot 9.
2. No arrival in time slot 8, arrival in time slot 9.
In scenario 1, the arrival in time slot 8 will be followed by the arrival in time slot 10, making T=4.
In scenario 2, the arrival in time slot 9 will be followed by no arrival in time slot 10, making T=3.
The passerby has no additional information about which of the two scenarios is true. Thus, the probability of each scenario is 1/2.
Therefore, the probability that T=4 is (1/2) * (1/2) = 1/4.
For the expected value of T, we need to calculate the weighted average of all possible values of T.
In scenario 1, T=4, with a probability of 1/4.
In scenario 2, T=3, with a probability of 1/4.
The expected value of T is calculated as:
Expected T = (4 * 1/4) + (3 * 1/4) = 4/4 = 1.
Therefore, the expected value of T is 1.
To find the probability that T=4, we need to consider the possible scenarios for the arrival in time slots 8 and 9. There are two possibilities:
1) The arrival occurred in time slot 8 and no arrival occurred in time slot 9. This would result in T=2.
2) No arrival occurred in time slot 8 and the arrival occurred in time slot 9. This would also result in T=2.
Therefore, the probability that T=4 is the sum of the probabilities of these two scenarios.
Let p be the probability of an arrival in any given time slot.
Scenario 1:
The probability of an arrival in time slot 8 is p.
The probability of no arrival in time slot 9 is 1-p.
Therefore, the probability of this scenario is p*(1-p).
Scenario 2:
The probability of no arrival in time slot 8 is 1-p.
The probability of an arrival in time slot 9 is p.
Therefore, the probability of this scenario is (1-p)*p.
The probability that T=4 is the sum of these two probabilities:
P(T=4) = p*(1-p) + (1-p)*p = 2p(1-p)
To find the expected value of T, we need to consider all possible values of T and their associated probabilities.
Possible values of T are 2, 3, 4, 5, 6, ...
The probability of T=2 is p*(1-p) + (1-p)*p = 2p(1-p) (as shown above).
For T=3:
The probability of an arrival in both time slots 8 and 9 is p*p = p^2.
Therefore, the probability of T=3 is p^2.
For T=4:
The probability of T=4 is 2p(1-p) (as shown above).
For T=5:
The probability of an arrival in time slot 8, no arrival in time slot 9, and an arrival in time slot 10 is p*(1-p)*p = p(1-p)^2.
The probability of no arrival in time slot 8, an arrival in time slot 9, and an arrival in time slot 10 is (1-p)*p*p = (1-p)p^2.
Therefore, the probability of T=5 is p(1-p)^2 + (1-p)p^2 = p(1-p)(1+p).
For T=6:
The probability of an arrival in time slots 8, 9, and 10 is p*p*p = p^3.
The probability of no arrival in time slot 8, an arrival in time slot 9, no arrival in time slot 10, and an arrival in time slot 11 is (1-p)*p*(1-p)*p = (1-p)^2p^2.
Therefore, the probability of T=6 is p^3 + (1-p)^2p^2 = p^2(1+p)(1-p).
We can continue this pattern for higher values of T, but it is clear that the probabilities will decrease as T increases.
Therefore, to calculate the expected value of T, we need to calculate the weighted average of all possible values of T.
Expected value of T = 2*(2p(1-p)) + 3*(p^2) + 4*(2p(1-p)) + 5*(p(1-p)(1+p)) + 6*(p^2(1+p)(1-p)) + ...
Simplifying this expression would require additional information about the probability distribution of the arrival in each time slot, which has not been provided in the problem statement.