Problem 28
Question
The law firm of Alam, Bartolini, Chinn, Dickinson, and Ellsberg has two senior partners: Alam and Bartolini. Two of the attorneys are to be selected to attend a conference. Assuming that all are equally likely to be selected, find each probability. (a) Chinn is selected. (b) Alam and Dickinson are selected. (c) At least one senior partner is selected.
Step-by-Step Solution
Verified Answer
(a) 0.4, (b) 0.1, (c) 0.7
1Step 1: Understanding the Basic Setup
There are five attorneys: Alam, Bartolini, Chinn, Dickinson, and Ellsberg. We are selecting two attorneys, and each attorney has an equal probability of being selected.
2Step 2: Total Possible Combinations
Calculate the total number of ways to select 2 attorneys from the 5 available. Use the combination formula: \[ \binom{n}{r} = \frac{n!}{r! (n-r)!} \] where \( n = 5 \) and \( r = 2 \). This gives: \[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \] There are 10 possible pairs of attorneys that can be selected.
3Step 3: Probability that Chinn is Selected
Find the number of combinations that include Chinn. If Chinn is to be selected, one more attorney from the remaining four (Alam, Bartolini, Dickinson, Ellsberg) is selected. Use the combination formula:\[ \binom{4}{1} = 4 \]Thus, there are 4 combinations including Chinn: (Chinn, Alam), (Chinn, Bartolini), (Chinn, Dickinson), and (Chinn, Ellsberg). The probability is: \[ \frac{4}{10} = 0.4 \]
4Step 4: Probability that Alam and Dickinson are Selected
There is only one way to select both Alam and Dickinson together, which is the pair (Alam, Dickinson). Hence, the probability is: \[ \frac{1}{10} = 0.1 \]
5Step 5: Probability that At Least One Senior Partner is Selected
List the pairs without any senior partners (only Chinn, Dickinson, and Ellsberg can form pairs without the seniors): - (Chinn, Dickinson)- (Chinn, Ellsberg)- (Dickinson, Ellsberg)There are 3 such pairs, so there are 7 pairs with at least one senior partner:\[ \binom{5}{2} - \text{{combinations without senior partners}} = 10 - 3 = 7 \]The probability is: \[ \frac{7}{10} = 0.7 \]
Key Concepts
CombinatoricsMathematical NotationProbability Calculations
Combinatorics
Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of elements within a set. It's crucial for solving problems where we need to determine how many possible ways something can occur. In probability, combinatorics helps us calculate the number of favorable outcomes relative to the total number of possible outcomes.
To solve our given problem, we use the concept of combinations because the order in which attorneys are selected does not matter. The formula for a combination is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the number of items to choose from, and \( r \) is the number of items to select.
This formula allows us to calculate all possible combinations of selecting 2 attorneys out of 5, which results in 10 combinations for our exercise. Understanding how to use and apply this formula is key in the field of probability and statistics.
To solve our given problem, we use the concept of combinations because the order in which attorneys are selected does not matter. The formula for a combination is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the number of items to choose from, and \( r \) is the number of items to select.
This formula allows us to calculate all possible combinations of selecting 2 attorneys out of 5, which results in 10 combinations for our exercise. Understanding how to use and apply this formula is key in the field of probability and statistics.
Mathematical Notation
Mathematical notation provides a systematic way of expressing mathematical concepts clearly and succinctly. In probability and statistics, notations such as \( \binom{n}{r} \) represent combinations, while factorial notation \( n! \) indicates the product of all positive integers up to \( n \).
Notation plays a vital role in forming and communicating mathematical ideas effectively. For example, within the exercise, using \( \binom{4}{1} \) efficiently tells us how many ways we can select the remaining attorney once Chinn is chosen.
Learning to interpret and apply these symbols is essential for students to grasp complex mathematical ideas as they provide a universal language that makes problem-solving more straightforward.
Notation plays a vital role in forming and communicating mathematical ideas effectively. For example, within the exercise, using \( \binom{4}{1} \) efficiently tells us how many ways we can select the remaining attorney once Chinn is chosen.
Learning to interpret and apply these symbols is essential for students to grasp complex mathematical ideas as they provide a universal language that makes problem-solving more straightforward.
Probability Calculations
Probability calculations enable us to determine the likelihood of a specific event happening. It's defined as the number of favorable outcomes divided by the total number of possible outcomes.
In the exercise, we calculate probabilities using our results from combinatorics. For example, the probability of Chinn being selected, among others, is derived from dividing the 4 favorable combinations by the total 10 combinations, thus giving us a probability of 0.4.
Similarly, by determining specific pairings such as Alam and Dickinson, we identify there's only one such favorable outcome, leading to a probability calculation of 0.1. Lastly, when assessing the selection of at least one senior partner, the exclusion of non-senior combinations helps in swiftly computing the desired probability, yielding 0.7.
In the exercise, we calculate probabilities using our results from combinatorics. For example, the probability of Chinn being selected, among others, is derived from dividing the 4 favorable combinations by the total 10 combinations, thus giving us a probability of 0.4.
Similarly, by determining specific pairings such as Alam and Dickinson, we identify there's only one such favorable outcome, leading to a probability calculation of 0.1. Lastly, when assessing the selection of at least one senior partner, the exclusion of non-senior combinations helps in swiftly computing the desired probability, yielding 0.7.
- Step 1: Identify total outcomes (using combinatorics).
- Step 2: Determine favorable outcomes.
- Step 3: Calculate probability by dividing favorable outcomes by total outcomes.
Other exercises in this chapter
Problem 27
Find the first four terms of each sequence. $$a_{1}=2, a_{2}=3, a_{n}=a_{n-1} \cdot a_{n-2} \text { for } n>2$$
View solution Problem 28
Find \(a_{1}\) for each arithmetic sequence. $$a_{6}=-8, a_{7}=-18$$
View solution Problem 28
Write the binomial expansion for each expression. $$\left(m+n^{2}\right)^{4}$$
View solution Problem 28
Prove each statement by mathematical induction. $$\text { If } a>1, \text { then } a^{n}>a^{n-1}$$
View solution