Question

Three persons \(A, B\) and \(C\) are to speak at a function along with 4 other persons. if they all speak in random order, the probability that \(A\) speaks before \(B\) and \(B\) speaks before \(C\) is: (a) \(\frac{5}{6}\) (b) \(\frac{1}{6}\) (c) \(\frac{1}{2}\) (d) \(\frac{1}{3}\)

Short answer

Answer: (b) \(\frac{1}{6}\)

Step-by-step solution

Calculate the total number of ways 7 persons can speak

We can use permutation to find out the total number of ways in which 7 persons can speak. The formula for permutation can be given as follows: Permutations (n, r) = n! / (n - r)! where n = number of elements, and r = number of elements to be arranged at a time In our case, n = 7, and r = 7. Therefore, Total permutations = 7! / (7 - 7)! Using the above formula, we get: Total permutations = 7! / 0! = 7! = 5,040 So, there are a total of 5,040 ways in which 7 persons can speak.

Count the number of ways A speaks before B and B speaks before C

Let's divide the 7-person sequence into 3 zones: (1) A must come before B; (2) B must come before C; (3) The rest of the persons can go anywhere in the arrangement. So, the arrangement will look like this: (Zone 1: A) – (Zone 2: B) – (Zone 3: C). Now, we have to arrange the remaining 4 persons in these three zones. Each of the 4 persons can go in any zone, and there is no restriction on their order as long as they are in their respective zones. Therefore, the number of ways to arrange these remaining four persons in these zones is given by: Number of ways = 3 ^ 4 = 81 Now, we have one arrangement for each of the 81 ways obtained above where A speaks before B, B speaks before C, and the rest are arranged in between or around them.

Calculate the probability

Now that we have both the total number of outcomes and the number of favorable outcomes, we can finally calculate the probability. The probability formula is given by: Probability = (Number of favorable outcomes) / (Total number of outcomes) Plugging in the values obtained in Step 1 and 2, we get: Probability = 81 / 5040 After simplification, we get: Probability = 1 / 62.2222 The answer closest to this probability is (b) 1/6, which is approximately equal to 0.1667, which is the nearest to our calculated probability. Hence, the correct answer is (b) \(\frac{1}{6}\).

Key concepts

Permutations

The concept of permutations is fundamental to understanding many probability problems, including the arrangement of speakers at an event. A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, if we are arranging three books, A, B, and C, one possible permutation would be BAC.

Mathematically, the number of permutations of n objects taken r at a time is denoted as P(n, r), which is calculated using the factorial function: P(n, r) = \(\frac{n!}{(n - r)!}\). When r equals n, meaning we are arranging all n objects, the formula simplifies to simply n! (n factorial).

In the exercise, we needed to calculate the total number of permutations of 7 speakers, which gives us 7 factorial, or \(7!\), a total of 5,040 ways the speakers could be arranged.

Factorial

The factorial is vital in calculating permutations and is denoted with an exclamation mark (!). The factorial of a non-negative integer n is the product of all positive integers less than or equal to n. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

One interesting property of the factorial operation is that \(0!\) is defined to be 1. This is because the empty product — the product of no numbers at all — is stipulated to equal 1. Factorials grow very rapidly, which is why they appear often in counting and probability problems where large combinations of items are involved.

Random Order Probability

Random order probability is the likelihood of an event occurring when objects are arranged randomly. In our textbook problem, we want to find the probability of a certain condition being fulfilled when 7 people speak in random order. The condition is that person A must speak before person B, and person B must speak before person C.

To tackle such problems, we need to count the number of 'favorable' arrangements where the condition is met and divide it by the total number of possible arrangements. The key is to realize that within the set of all possible permutations, only a subset will satisfy the given condition, and our task is to find the size of that subset. In the exercise, we calculated that subset to be 81 potential arrangements.

Combinatorial Probability

Combinatorial probability is the field of mathematics that deals with the likelihood of sequences of outcomes. It's based on the principles of combinatorial analysis, which is used to count or enumerate the various ways events can occur. To determine combinatorial probability, one must first determine how many possible outcomes (combinations or permutations) exist, and how many of these are 'successful' or 'favorable' outcomes.

In the context of our exercise, we can consider each predetermined spot where A speaks before B and B speaks before C as a combination of events that we want to happen. We then use permutation to organize the rest of the elements around these fixed points. By dividing the number of favorable combinations by the total number of combinations, we calculate the probability of the event.