boys and 10 girls decide to rent a 13-passenger van and a 5-passenger car to drive to a concert in a nearby city. If the group is distributed randomly between the vehicles, what is the probability that
Either Tom or Sarah in the van (but not both)?
This is TERRIBLY incomplete! Care to type it correctly?
8 boys and 10 girls decide to rent a 13-passenger van and a 5-passenger car to drive to a concert in a nearby city. If the group is distributed randomly between the vehicles, what is the probability that either Tom(a boy) or Sarah (a girl) is in the van?
To find the probability that either Tom or Sarah is in the van (but not both), we need to compute the probability of each case separately and then sum them up.
Let's start by calculating the probability of Tom being in the van while Sarah is not.
Step 1: Determine the total number of ways to distribute the group between the two vehicles.
We have 10 girls, so there are 10! (factorial) ways to arrange them in the van. Likewise, there are 10! ways to arrange them in the car. Therefore, there are (10!)(10!) total ways to distribute the group between the two vehicles.
Step 2: Determine the number of ways to distribute the group such that Tom is in the van while Sarah is not.
Since Tom is a single individual, he can only be in either the van or the car. Therefore, there are 2 ways to assign Tom to one of the vehicles.
For the remaining 11 individuals (10 girls and Sarah), there are (11!) ways to arrange them between the van and the car.
Step 3: Calculate the probability.
The probability of Tom being in the van while Sarah is not is given by:
(2 * 11!) / [(10!) * (10!)]
Next, let's calculate the probability of Sarah being in the van while Tom is not.
Step 1: Determine the total number of ways to distribute the group between the two vehicles.
The total number of ways to distribute the group is the same as calculated before, which is (10!)(10!).
Step 2: Determine the number of ways to distribute the group such that Sarah is in the van while Tom is not.
Since Sarah is a single individual, she can only be in either the van or the car. Therefore, there are 2 ways to assign Sarah to one of the vehicles.
For the remaining 11 individuals (10 girls and Tom), there are (11!) ways to arrange them between the van and the car.
Step 3: Calculate the probability.
The probability of Sarah being in the van while Tom is not is given by:
(2 * 11!) / [(10!) * (10!)]
Finally, we add the probabilities from both cases:
(2 * 11!) / [(10!) * (10!)] + (2 * 11!) / [(10!) * (10!)]
Simplifying this expression will give us the final probability.
To find the probability that either Tom or Sarah is in the van (but not both), we need to calculate the probability of two mutually exclusive events: Tom being in the van and Sarah not being in the van, and Sarah being in the van and Tom not being in the van.
Let's start by determining the number of ways we can distribute the boys and girls into the vehicles. Since there are 13 seats in the van and 5 seats in the car, we have:
- For the van: 13 seats and 10 girls, so we have (13 choose 1) * (10 choose 0) = 13 ways to select the boys and girls for the van.
- For the car: 5 seats and 10 girls, so we have (5 choose 0) * (10 choose 10) = 1 way to select the boys and girls for the car.
Now, let's consider the cases in which either Tom or Sarah is in the van, but not both:
- Tom is in the van, and Sarah is not: We need to select 1 seat for Tom from the 13 seats in the van, and select all 10 girls from the remaining available seats (12 seats are left). So we have (13 choose 1) * (12 choose 10) = 13 * 66 = 858 ways.
- Sarah is in the van, and Tom is not: We need to select 1 seat for Sarah from the 13 seats in the van, and select all 10 girls from the remaining available seats (12 seats are left). So we have (13 choose 1) * (12 choose 10) = 13 * 66 = 858 ways.
Since these two events are mutually exclusive (Tom being in the van and Sarah not being in the van), and (Sarah being in the van and Tom not being in the van), we can simply add up the number of ways for each event: 858 + 858 = 1716 ways.
Finally, the probability that either Tom or Sarah is in the van (but not both) is given by the number of favorable outcomes (1716 ways) divided by the total number of possible outcomes (13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) since both vehicles are completely filled:
Probability = (1716) / (13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1).
Evaluating the expression, we get the probability = 0.037, or approximately 3.7%.
Therefore, the probability that either Tom or Sarah is in the van (but not both) is approximately 3.7%.