Problem 4 - Thinking Mathematically

Question

Seven performers, A, B, C, D, E, F, and G, are to appear in a fund raiser. The order of performance is determined by random selection. Find the probability that a. D will perform first. b. E will perform sixth and \(B\) will perform last. c. They will perform in the following order: C, D, B, A, G, \(\mathrm{F}, \mathrm{E}\). d. \(F\) or \(G\) will perform first.

Step-by-Step Solution

Answer

The solutions for the respective parts are: a. The probability that D performs first is 1/7. b. The probability that E performs sixth and B performs last is 1/42. c. The probability that they will perform in the order C, D, B, A, G, F, E is 1/7! (=1/5040). d. The probability that either F or G will perform first is 2/7.

Step 1: Calculate the straightforward probability

For part a, we just need to determine the probability that D will perform first. The probability of any one performer going first is given by 1 divided by the total number of performers. Therefore, the probability that D performs first is 1/7.

Step 2: Calculate the probability of specific individuals in specific places

For part b, we are trying to determine the probability that E will perform sixth and B will perform last. Here, we need to consider three independent events: E performing 6th, B performing 7th, and the placement of the remaining 5 performers. The probability of each event is 1/7, 1/6 and 5!/5! = 1 respectively, and multiplying these together will give us our overall probability, which is (1/7)*(1/6)*1 = 1/42.

Step 3: Calculate the probability of a specific order

For part c, they are asking for the probability of the performers appearing in one specific order. Since there is only one order that matches this out of 7!, the probability of this would be 1/7!

Step 4: Calculate the probability of one of two events

For part d, the problem is asking for the probability that either F or G will perform first. The 'or' in this problem means we are looking for the probability of F performing first plus the probability of G performing first. Given that the probability of any one performer going first is 1/7, we simply add these two probabilities together; so our answer is 1/7 + 1/7 = 2/7.

Key Concepts

Random Selection

When we talk about random selection, we are referring to the process where each subject has an equal chance to be chosen. In the context of our exercise with performers A through G, random selection is key to determining who goes on stage first.

• There are 7 performers, each with an equal probability of being selected.
• The probability that a specific performer, let's say Performer D, gets selected to perform first is calculated by dividing 1 by the total number of performers, which is 7. That gives us a probability of \( \frac{1}{7} \).

This approach of assigning each performer the same chance ensures that the selection process is fair and unbiased. It's the foundational concept in probability when the elements are equitably considered.

Order of Performance

Understanding the order of performance is essential when dealing with permutations. In scenarios where we care about the specific sequence or order of events, we delve into permutations. The order of performance matters when instructions demand that particular events occur in a precise sequence like in our exercise.

• For example, if we want the performers to appear in the order C, D, B, A, G, F, E, we have only one unique sequence we are interested in.
• The probability of them appearing exactly in that order is calculated by recognizing there's only one such sequence out of all possible permutations (7!), making the probability \( \frac{1}{7!} \).

Permutations aren't concerned with choosing individual elements but with arranging all of them in a specified sequence. It's crucial to grasp this concept to solve problems involving specific ordering.

Independent Events

In probability, independent events refer to scenarios where the outcome of one event does not affect the outcome of another. In the exercise concerning performers E and B, this concept is particularly useful.

• We want to find out the probability of E performing 6th and B performing last. These are independent events because the placement of E does not influence B's position.
• By considering each of these events separately, we assign probabilities like \( \frac{1}{7} \) for E being in the 6th position and \( \frac{1}{6} \) for B being in the last position.
• The compound probability for both occurring in that specific manner is the product of their individual probabilities: \( \frac{1}{7} \times \frac{1}{6} = \frac{1}{42} \).

Understanding independent events helps us evaluate probabilities for complex situations by simplifying them into manageable parts.