a We want to find the probability that you will be the last performer and your rival performs immediately before you. To find it we will use the theoretical probability.
P=Favorable Outcomes/Possible Outcomes
We will start by finding the number of possible outcomes, which is the number of permutations of 10 students performing in a school talent show. We use permutations because the order in which students perform is important. Let's recall the formula for the number of permutations of n objects.
_nP_n=n!
In our case, there are a total of 10 students performing, n=10. Therefore, the number of possible outcomes is 10!. Next, we will look for the number of favorable outcomes. The intermission occurs after the 5th act. We want you to be the last performer before intermission and your rival performs immediately before you.
Your position and your rival's position are fixed. We need to take into account only the order of the other performers.
10-1-1= 8
There are 8 students whose order can be determined at random. Therefore, we will calculate the number of permutations of 8 other students.
_8P_8= 8!
We found that the number of favorable outcomes is 8!. This means that we have enough information to calculate the desired probability.
The probability that you will be the last performer before intermission and your rival performs immediately before you is equal to 190.
b
We want to find the probability that you will not be the first performer. In order to solve this problem, we need to consider the complement of the event we are interested in.
A: you are not the first performer
A: you are the first performer
If we find the probability of event A we will be able to obtain the probability of event A, since they are related by the Probability of the Complement of the Event Formula.
The probability of the complement of event A is P(A) = 1 - P(A).
From now on, we will consider the event A. To find P(A), we will use the theoretical probability.
P=Favorable Outcomes/Possible Outcomes
We will start by obtaining the number of possible outcomes. As in Part A, the order in which the students perform is important. Therefore, the number of possible outcomes is again equal to 10!. Let's look for the number of favorable outcomes. We want you to be the first performer.
Your position is fixed. Therefore, we need to take into account only the order of the other performers.
10-1= 9
There are 9 students whose order can be determined at random. Therefore, we will calculate the number of permutations of 9 other students.
_9P_9= 9!
We found that the number of favorable outcomes is 9!. We are ready to find the probability P(A).
In order to find the probability of Event A that you are not the first performer, we have to substitute the obtained probability into the formula for the probability of the complement of event.
