Figures obtained from a city's police department seem to indicate that of all motor vehicles reported as stolen, 61% were stolen by professionals (P), whereas 39% were stolen by amateurs (primarily for joy rides) (A). Of those vehicles stolen by professionals, 21% were recovered within 48 hours, 25% were recovered after 48 hours, and 54% were never recovered. Of those vehicles presumed stolen by amateurs, 30% were recovered within 48 hours, 64% were recovered after 48 hours, and 6% were never recovered.
(a) Draw a tree diagram representing the data.
(b)What is the probability that a vehicle stolen by a professional in this city will be recovered within 48 hours? (Round your answer to four decimal places.)
(c)What is the probability that a vehicle stolen in this city will never be recovered? (Round your answer to four decimal places.)
To answer these questions, we can use the tree diagram method to visualize the different possibilities and calculate the probabilities. Let's go step by step:
(a) First, let's draw the tree diagram representing the data.
```
Stolen by Professionals (P)
/ \
Recovered within 48 hours Recovered after 48 hours Never recovered
(21%) (25%) (54%)
/
/
Stolen by Amateurs (A)
/
/
Recovered within 48 hours Recovered after 48 hours Never recovered
(30%) (64%) (6%)
```
(b) Now, let's calculate the probability that a vehicle stolen by a professional in this city will be recovered within 48 hours.
To calculate this, we need to find the probability of the path that leads to a vehicle stolen by professionals being recovered within 48 hours.
The probability of a vehicle being stolen by professionals and being recovered within 48 hours is:
P(P ∩ Recovered within 48 hours) = P(P) * P(Recovered within 48 hours given P)
= 0.61 * 0.21
= 0.1281 (rounded to four decimal places)
So, the probability that a vehicle stolen by a professional in this city will be recovered within 48 hours is approximately 0.1281.
(c) Next, let's determine the probability that a vehicle stolen in this city will never be recovered.
To calculate this, we need to sum up the probabilities of both paths where the vehicle is never recovered: stolen by professionals and stolen by amateurs.
The probability of a vehicle being stolen by professionals and never being recovered is:
P(P ∩ Never recovered) = P(P) * P(Never recovered given P)
= 0.61 * 0.54
= 0.3294
The probability of a vehicle being stolen by amateurs and never being recovered is:
P(A ∩ Never recovered) = P(A) * P(Never recovered given A)
= 0.39 * 0.06
= 0.0234
Now, to find the overall probability that a vehicle stolen in this city will never be recovered, we add these two probabilities:
P(Never recovered) = P(P ∩ Never recovered) + P(A ∩ Never recovered)
= 0.3294 + 0.0234
= 0.3528 (rounded to four decimal places)
So, the probability that a vehicle stolen in this city will never be recovered is approximately 0.3528.
(a) The tree diagram representing the data is as follows:
```
P (61%)
/ \
recovered within 48 hours (21%) recovered after 48 hours (25%) never recovered (54%)
A (39%)
/ \
recovered within 48 hours (30%) recovered after 48 hours (64%) never recovered (6%)
```
(b) To find the probability that a vehicle stolen by a professional in this city will be recovered within 48 hours, we multiply the probabilities along the path leading to "recovered within 48 hours" for professionals.
P(recovered within 48 hours for professionals) = P(P) * P(recovered within 48 hours | P) = 0.61 * 0.21 = 0.1281
Therefore, the probability that a vehicle stolen by a professional in this city will be recovered within 48 hours is 0.1281.
(c) To find the probability that a vehicle stolen in this city will never be recovered, we add the probabilities of vehicles stolen by professionals and vehicles stolen by amateurs that are never recovered.
P(never recovered) = P(P) * P(never recovered | P) + P(A) * P(never recovered | A) = 0.61 * 0.54 + 0.39 * 0.06 = 0.3486 + 0.0234 = 0.3720
Therefore, the probability that a vehicle stolen in this city will never be recovered is 0.3720.