Assume that there are 16 board members: 10 females, and 6 males including Larry. There are 4 tasks to be assigned. Note that assigning the same people different tasks constitutes a different assignment.

(1) Find the probability that both males and females are given a task.

(2) Find the probability that Larry and at least one female are given tasks.

(1) That would be 1-(prob. of no males)- (prob. of no females).

1 - (10*9*8*7)/(16*15*14*13)
- (6*5*4*3)/(16*15*14*13)
= 1 - (5040 + 360)/43680
= 1 - 5400/43680 = 1-135/1092
= 957/1092 = 319/364

2) Larry will be picked in
1 - (15/16)*(14/15)*(13*14)*(12/13)
= 1 - 32760/43680 = 1- 819/1092
= 273/1092 of the situations.
Next, calculate the fraction of those situations for which the three other tasks are assigned to one or more women. Multiply 273/1092 by that factor.

To solve these probability questions, we need to count the favorable outcomes and total possible outcomes.

(1) Find the probability that both males and females are given a task.

To solve this, we can calculate the probability of assigning tasks to all board members and subtract the probability of assigning tasks to only one gender.

Total Possible Outcomes:
For each task, there are 16 members to choose from, so the total possible outcomes for assigning 4 tasks are 16 * 16 * 16 * 16.

Favorable Outcomes - Assigning tasks to both males and females:
For each task, there are 6 males and 10 females to choose from.
So the favorable outcomes for assigning tasks to both males and females are 6 * 10 * 6 * 10.

Now, we can calculate the probability:

P(Both males and females are given a task) = Favorable outcomes / Total possible outcomes
= (6 * 10 * 6 * 10) / (16 * 16 * 16 * 16)

(2) Find the probability that Larry and at least one female are given tasks.

To solve this, we need to count the favorable outcomes of assigning tasks to Larry and at least one female out of the 10 available.

Total Possible Outcomes:
Again, for each task, there are 16 members to choose from, so the total possible outcomes for assigning 4 tasks are 16 * 16 * 16 * 16.

Favorable Outcomes - Assigning tasks to Larry and at least one female:
Larry has to be assigned a task, which leaves us with 3 tasks to assign to the remaining 15 members (10 females and 5 other members).
So the favorable outcomes can be calculated as (number of ways to assign tasks to Larry and at least one female) * (number of ways to assign the remaining tasks to other members).

Number of ways to assign tasks to Larry and at least one female = 1 (because Larry has to be assigned a task)
Number of ways to assign the remaining tasks to other members = 15 * 15 * 15

So the favorable outcomes are 1 * (15 * 15 * 15).

Now, we can calculate the probability:

P(Larry and at least one female are given tasks) = Favorable outcomes / Total possible outcomes
= (1 * (15 * 15 * 15)) / (16 * 16 * 16 * 16)

To find the probability for each scenario, we need to calculate the total number of possible assignments and the number of favorable assignments for each case.

(1) To find the probability that both males and females are given a task, we need to calculate the number of ways we can assign tasks to both males and females.

The total number of assignments can be calculated using the formula for combinations:

Number of ways to choose m out of n objects = n! / (m! * (n-m)!), where n is the total number of objects and m is the number of objects to be chosen.

We have 16 board members, and we are assigning tasks to all of them, so n = 16. We have 6 males, and we want to assign 4 tasks to them, so m = 6. Similarly, we have 10 females, and we want to assign 4 tasks to them as well, so m = 10.

The total number of assignments (favorable and non-favorable) would be the product of the number of ways to assign tasks to males and females:

Total number of assignments = (6! / (4! * (6-4)!)) * (10! / (4! * (10-4)!))

Now, to find the number of favorable assignments, we assume that all tasks are identical and there are no preferences among them. This means we just need to count the number of ways we can assign tasks to males and females, regardless of which specific task they receive.

Number of favorable assignments = (2^4) * (2^4)

The power of 2 comes from the fact that each task can either go to a male or a female, resulting in 2 possibilities.

Finally, we can calculate the probability by dividing the number of favorable assignments by the total number of assignments:

Probability = Number of favorable assignments / Total number of assignments

(2) To find the probability that Larry and at least one female are given tasks, we need to consider two cases:

a) Larry is given a task, and at least one female is assigned a task.
b) Larry is given a task, but no female board member is assigned a task.

To calculate the probability for case a, we can assign tasks to Larry, any female, and then the remaining tasks to any of the remaining board members.

Number of favorable assignments for case a = 1 (for Larry) * (10! / ((10-1)!)) * (15! / ((15-3)!))

To calculate the probability for case b, we can assign tasks to Larry, and then assign the remaining tasks to any of the remaining 15 board members who are not female.

Number of favorable assignments for case b = 1 (for Larry) * (15! / ((15-4)!))

The total number of favorable assignments would be the sum of the favorable assignments for both cases:

Total number of favorable assignments = Number of favorable assignments for case a + Number of favorable assignments for case b

Finally, we can calculate the probability by dividing the total number of favorable assignments by the total number of assignments (same as in part 1):

Probability = Total number of favorable assignments / Total number of assignments