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Chapter 10: Problem 12

There are 20 students in a class, and every day the teacher randomly selects 6 students to present a homework problem. Noah and Rita wonder what the chance is that they will both present a homework problem on the same day. a. How many different ways are there of selecting a group of 6 students? b. How many of these groups include both Noah and Rita? c. What is the probability that Noah and Rita will both be called on to give their reports?

Short Answer

Expert verified
The probability that Noah and Rita both present on the same day is \( \frac{1}{12} \).

Step by step solution

01

Total Ways to Select 6 Students

To find the total ways to select 6 students from 20, we use the combination formula \( \binom{n}{r} \), where \( n \) is the total number of students, and \( r \) is the number of students to choose. Thus, \( \binom{20}{6} \) calculates to:\[ \binom{20}{6} = \frac{20!}{6!(20-6)!} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 38760. \] There are 38,760 ways to select 6 students.
02

Ways to Select a Group Including Noah and Rita

If Noah and Rita are included in the group, we need to select 4 more students from the remaining 18 students. We use \( \binom{18}{4} \) to find this number:\[ \binom{18}{4} = \frac{18!}{4!(18-4)!} = \frac{18 \times 17 \times 16 \times 15}{4 \times 3 \times 2 \times 1} = 3060. \] There are 3,060 ways to form a group of 6 that includes both Noah and Rita.
03

Calculate Probability Noah and Rita are Both Called

The probability that both Noah and Rita are selected is the ratio of the favorable outcomes (groups that include Noah and Rita) to the total outcomes (all groups of 6 students). The probability \( P \) is calculated as:\[ P = \frac{\text{Number of groups including Noah and Rita}}{\text{Total number of groups}} = \frac{3060}{38760} = \frac{1}{12}. \] Hence, the probability that Noah and Rita will both present on the same day is \( \frac{1}{12} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics that deals with counting and arranging objects. It's the science of determining the number of possible outcomes in different situations. For example, when you have a group of items and you want to know how many different ways you can select or arrange a specific number of them, combinatorics comes into play. In the given exercise, it refers to how we can form groups from a specific number of students. It involves using a formula to calculate combinations to determine all the possible selections. This is especially useful when you need to consider combinations without caring about the order - such as choosing 6 students out of a class of 20.
Probability Calculation
Probability calculation is all about determining how likely an event is to occur. It helps quantify uncertainty by expressing chances as numbers between 0 and 1. A probability of 0 means the event definitely will not happen, and a probability of 1 means it definitely will. In our exercise example, we had to find the probability that Noah and Rita are both chosen to present their homework. We calculated this by finding two key pieces of information:
  • The total number of ways to select any 6 students from 20.
  • The number of ways to select 6 students including both Noah and Rita.
Once we have these numbers, the probability is expressed as a fraction of the number of favorable outcomes over the total number of outcomes. This fraction tells us just how likely it is that Noah and Rita will present together on the same day.
Combination Formula
The combination formula is used to calculate the number of ways to select a certain number of items from a larger pool, where the order doesn't matter. It's represented mathematically with the notation \( \binom{n}{r} \) where \( n \) is the total number of items, and \( r \) is the number of items to choose. The formula is structured as follows:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!}\]Here, \(!\) denotes a factorial, which means multiplying a series of descending natural numbers. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).In our exercise, we used this formula to compute:
  • \( \binom{20}{6} \) to find the total combinations of selecting 6 students from 20.
  • \( \binom{18}{4} \) to find combinations where Noah and Rita are already included, leaving 18 students to choose the remaining 4 students from.
By applying this formula, we discover all possible groupings, which helps in determining probabilities of certain groupings occurring.

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Most popular questions from this chapter

APPLICATION Suppose there are 180 twelfth graders in your school, and the school records show that 74 of them will be attending college outside their home state. You conduct a survey of 50 twelfth graders, and 15 tell you that they will be leaving the state to attend college. What is the theoretical probability that a random twelfth grader will be leaving the state to attend college? Based on your survey results, what is the experimental probability? What could explain the difference? (a)

Arrange the following events from most likely to least likely. Explain your thinking a. being born right-handed b. flipping a single head if you flip a coin four times c. not watching any television this week d. being taller than the average class height e. no one being late to your math class this week

Create a tree diagram with probabilities showing outcomes when drawing two marbles with replacement from a bag containing one blue and two red marbles. (You do replace the first marble drawn from the bag before drawing the second.)

Evaluate each number of permutations or combinations without using your calculator. Show your calculations. a. \({ }_{5} P_{3}\) (a) b. \({ }_{5} C_{3}\) c. \({ }_{5} P_{4}\) d. \({ }_{5} C_{4}\)

You have purchased 4 tickets to a school music department raffle. Three prizes will be awarded, and 150 tickets were sold. a. How many ways can the three prizes be assigned to the 150 tickets if the prizes are different? (a) b. How many ways can the three prizes be assigned to the 150 tickets if the prizes are the same?
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