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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 will both present on the same day is approximately 0.0789.

Step by step solution

01

Calculate Total Ways to Select 6 Students

To find the total number of ways to choose 6 students from 20, use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 20 \) and \( r = 6 \): \[ \binom{20}{6} = \frac{20!}{6! \, 14!} = 38760 \]Thus, there are 38,760 ways to select 6 students from a group of 20.
02

Calculate Ways to Select 4 More Students with Noah and Rita

If Noah and Rita are already selected, we need to choose 4 more students from the remaining 18 students (since two spots are filled by Noah and Rita). Use the combination formula:\[ \binom{18}{4} = \frac{18!}{4! \, 14!} = 3060 \]Thus, there are 3,060 ways to pick the remaining 4 students when Noah and Rita are already selected.
03

Calculate Probability That Both Noah and Rita Present

The probability that Noah and Rita are both selected is given by the ratio of the number of favorable outcomes to the total number of ways to choose the students:\[ \text{Probability} = \frac{\text{Number of ways to select 4 students with Noah and Rita}}{\text{Total ways to select 6 students}} = \frac{3060}{38760} \approx 0.0789 \]Therefore, the probability that both Noah and Rita are selected on the same day is approximately 0.0789.

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Key Concepts

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

Combination Formula
Understanding how to choose a subset of items from a larger set is essential in probability calculations. The combination formula helps us accomplish this by focusing on selecting items without regard to the order.
The formula to calculate combinations is given by:
\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]
Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to select. The exclamation mark represents a factorial, which means multiplying a series of descending natural numbers. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
  • Examples include choosing 3 colors out of 5 in a palette or selecting 4 team members out of a group of 10 for a task.
  • In the exercise, the combination formula helps calculate how many different groups of 6 students can be formed from a class of 20. This step is crucial for determining probabilities.
Combinatorics
Combinatorics is the branch of mathematics that deals with counting combinations and permutations of elements in sets. It goes beyond just the combination formula by analyzing different scenarios and counting methods.
In our exercise, combinatorics helps us not just determine the number of general groups but also those that meet specific criteria, such as including Noah and Rita.
  • This involves subtracting or modifying the available pool of elements. Once Noah and Rita are chosen, we aren't picking from the full group anymore. We focus on the remaining elements.
  • This is done by recalculating the combinations for the reduced pool, which is now 18 students instead of 20, since Noah and Rita have been pre-selected.
  • Combinatorics makes it easier to solve complex probability problems by breaking them down into simpler counts and logical analyses.
Statistics
Statistics is all about analyzing data and drawing conclusions. In problems involving probability, such as our exercise, it helps us understand the likelihood of various outcomes.
The probability that Noah and Rita are both chosen is a statistical question about the expected composition of our student group.
  • We start by determining the total number of possible outcomes, which in this case, are all possible groups of 6 students from a class of 20, as calculated using the combination formula.
  • Next, we identify the "favorable outcomes," which are the specific combinations where both Noah and Rita are included. Combinatorics aids in finding these counts efficiently.
  • The probability is then calculated by dividing the number of favorable outcomes by the total number of outcomes. This ratio reveals the likelihood of our event of interest occurring.
The use of statistics in probability involves comparing possible scenarios and calculating how likely certain scenarios are, which is fundamental in making informed predictions and decisions.

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Most popular questions from this chapter

At this point in the season, Jackson has made 35 out of his 50 free throw attempts, so he's been successful on \(70 \%\) of his free throws. He wants to improve his rate to \(80 \%\) as soon as possible. a. How many consecutive free throws must he make to reach this goal? (a) b. If his probability of making any one shot is \(70 \%\), what is the probability that he will perform the number of consecutive free throws you found in 12a? (a)

Create a tree diagram with probabilities showing outcomes when drawing two marbles without replacement from a bag containing one blue and two red marbles. (You do not replace the first marble drawn from the bag before drawing the second.) (a)

Cheryl plays on the school basketball team. When shooting free throws, she makes \(75 \%\) of her first shots, and \(80 \%\) of her second shots provided she makes the first one. However, if she misses the first shot, she makes only half of her second shots. Each free throw is worth one point. a. Draw a tree diagram of a two-shot attempt. What is the probability that she will make both shots? (a) b. What is the expected number of points that Cheryl will make in a two-shot free throw attempt? (a) c. If Cheryl has five chances to shoot two free throws in a game, how many points can she expect to make? (a)

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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.)
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