Assume that there are 17 board
members: 9 females, and 8 males including Carl. There are 4 tasks to be assigned.
What is the probability that at least one female and Carl are assigned tasks?
number of task assignments without restrictions = C(17,4) = 2380
Carl ---- C
Female --F
Male ----M
cases:
CFMM --- 1*C(9,1)*C(7,2) = 189
CFFM ----1*C(9,2)*C(7,1) = 252
CFFF -----1*C(9,3)*C(7,0) = 84
total cases with Carl and at least one female = 525
prob of the event as stated = 525/2380 = 15/68
To find the probability that at least one female and Carl are assigned tasks, we will calculate the probability of the complement event (i.e., the event that no female or Carl is assigned tasks) and subtract it from 1.
Step 1: Find the probability of no female or Carl being assigned tasks.
To calculate this probability, we will consider the number of ways we can assign the tasks to the remaining members who are not female or Carl.
Number of remaining members (excluding females and Carl) = Total members - Females - Carl = 17 - 9 - 1 = 7
Number of ways to select members from the remaining members = 7 choose 4 (denoted as C(7, 4))
Step 2: Find the total number of ways to assign the tasks.
To calculate this, we will consider the number of ways we can assign the tasks to all board members.
Total number of members = Total members = 17
Total number of ways to assign the tasks = 17 choose 4 (denoted as C(17, 4))
Step 3: Calculate the probability of no female or Carl being assigned tasks.
Probability of no female or Carl being assigned tasks = (Number of ways to select members from the remaining members) / (Total number of ways to assign the tasks)
= C(7, 4) / C(17, 4)
Step 4: Calculate the probability of at least one female and Carl being assigned tasks.
Probability of at least one female and Carl being assigned tasks = 1 - Probability of no female or Carl being assigned tasks
Now, let's calculate the values step by step.
Step 1: Calculate the number of ways to select members from the remaining members:
C(7, 4) = (7!) / (4!(7-4)!) = (7 * 6 * 5) / (3 * 2 * 1) = 35
Step 2: Calculate the total number of ways to assign the tasks:
C(17, 4) = (17!) / (4!(17-4)!) = (17 * 16 * 15 * 14) / (4 * 3 * 2 * 1) = 2380
Step 3: Calculate the probability of no female or Carl being assigned tasks:
Probability = 35 / 2380 = 0.0147 (rounded to 4 decimal places)
Step 4: Calculate the probability of at least one female and Carl being assigned tasks:
Probability = 1 - 0.0147 = 0.9853 (rounded to 4 decimal places)
Therefore, the probability that at least one female and Carl are assigned tasks is approximately 0.9853.
To answer this question, we need to find the probability of at least one female and Carl being assigned tasks out of a total of 17 board members (9 females and 8 males).
First, let's find the probability of Carl being assigned a task. Since there are four tasks to be assigned and a total of 17 board members, the probability of Carl being assigned a task is 4/17.
Next, let's find the probability of at least one female being assigned a task. To do this, we can find the probability of no female being assigned a task and subtract it from 1.
The probability of no female being assigned a task is calculated by selecting all the tasks (4) from only the male board members (8 males) divided by selecting all tasks (4) from all the board members (17 total). This can be calculated using the formula:
P(no female assigned) = C(4, 8) / C(4, 17)
Where C(n, r) represents the combination, which calculates the number of ways to choose r items from a set of n items. In this case, we want to choose 4 tasks from the 8 males and divide it by choosing 4 tasks from all 17 board members.
Once we have the probability of no female being assigned a task, we can find the probability of at least one female being assigned a task by subtracting this probability from 1. So,
P(at least one female assigned) = 1 - P(no female assigned)
Now, we can calculate the probabilities:
P(Carl assigned) = 4/17
P(no female assigned) = C(4, 8) / C(4, 17)
P(at least one female assigned) = 1 - P(no female assigned)
By calculating these probabilities, we can find the probability of at least one female and Carl being assigned tasks.