/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 At this point in the season, Jac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 12

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)

Short Answer

Expert verified
a) Jackson needs to make 25 consecutive free throws. (a) b) Probability of making 25 consecutive shots is approximately 0.00000899.

Step by step solution

01

Determine Current Success Rate

Jackson's current success rate for free throws is given as \(70\%\). He has made 35 successful throws out of 50 attempts. This aligns with the success rate calculation: \(\frac{35}{50} \times 100\% = 70\%\).
02

Set Up Equation for New Success Rate

We want to find the number of additional successful free throws Jackson needs to make in order for his success rate to reach \(80\%\). Let \(x\) be the number of consecutive successful free throws he needs to make. His total successful throws would then be \(35 + x\), and total attempts would be \(50 + x\). The equation becomes \(\frac{35 + x}{50 + x} = 0.8\).
03

Solve for x

Solve the equation \(\frac{35 + x}{50 + x} = 0.8\) to find \(x\). Multiply both sides of the equation by \(50 + x\) to clear the fraction: \(35 + x = 0.8 \times (50 + x)\). This simplifies to \(35 + x = 40 + 0.8x\). Subtract \(0.8x\) from both sides: \(35 + 0.2x = 40\). Then subtract 35 from both sides: \(0.2x = 5\). Finally, divide by 0.2: \(x = \frac{5}{0.2} = 25\). Jackson needs to make 25 consecutive free throws.
04

Calculate Probability of Success

The probability of Jackson making each free throw is \(70\%\) or \(0.7\). We need the probability of him making 25 consecutive free throws to achieve \(80\%\) success rate. The probability of making \(x\) consecutive free throws is given by \((0.7)^x\). Substitute \(x = 25\) into the formula: \((0.7)^{25}\). This evaluates to a probability of approximately \(0.00000899\) (using a calculator).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability
Probability is the mathematics of chance. It's about predicting the likelihood of an event happening. The probability of an event is a number between 0 and 1, where 0 means the event will not occur, and 1 means it will definitely occur. For example, if Jackson has a probability of 0.7, or 70%, of making a free throw, this means that for each shot he takes, there is a 70% chance it will go in the basket.

To calculate the probability of independent events happening in sequence, such as making several free throws in a row, you multiply the probability of each event. If each shot has a probability of 0.7, then the probability of Jackson making 2 consecutive shots is \(0.7 \times 0.7 = 0.49\). Similarly, the probability of making 25 consecutive shots is \((0.7)^{25}\) which is about 0.00000899. This shows that while Jackson could theoretically make 25 shots in a row, it's very unlikely.
Equation Solving
Equation solving is a primary tool in algebra. It's about finding the unknown value in a math sentence that describes a relationship. In Jackson's case, the equation \(\frac{35 + x}{50 + x} = 0.8\) helps determine how many more successful free throws he needs to achieve an 80% success rate.

To solve the equation, you first clear the fraction by multiplying both sides by \(50 + x\). This eliminates the denominator, simplifying to \(35 + x = 0.8 \times (50 + x)\). Expanding the right side, you get \(35 + x = 40 + 0.8x\). By rearranging terms through subtracting 0.8x from both sides, and then subtracting 35, you isolate x on one side: \(0.2x = 5\). Finally, dividing both sides by 0.2 reveals that \(x = 25\).
Success Rate
A success rate is the ratio of successful outcomes to total attempts, expressed as a percentage. It measures how often success occurs. Jackson's current success rate of making free throws is 70%.

To calculate success rate, use the formula: \(\frac{\text{successful attempts}}{\text{total attempts}} \times 100\%\). Jackson's aim is to increase his rate to 80% by making additional successful shots. This requires him to increase his current total of successful shots to align with a new target success rate.

Understanding how to manipulate success rates is useful in various contexts, such as improving sports performance, assessing work productivity, or even analyzing business outcomes. In Jackson's case, reaching his desired success rate involves accounting for both current performance and the path towards improvement.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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.)

Here are the results for 100 rolls of a six-sided die. \(\begin{array}{|l|r|r|r|r|r|r|} \hline \text { Number rolled } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Tally } & 16 & 15 & 19 & 18 & 14 & 18 \\ \hline \end{array}\) a. Based on the results of the experiment, what is the experimental probability of rolling a 6 ? b. What is the theoretical probability of rolling a 6 ? c. What is the experimental probability of rolling an even number? d. What is the theoretical probability of rolling an even number? e. Are the experimental and theoretical probabilities in \(13 \mathrm{a}-\mathrm{b}\) and \(13 \mathrm{c}-\mathrm{d}\) close to each other? Based on these results, do you think the die is fair?

In 2001 there were 3141 counties in the United States. Here are data on five counties: \(\begin{array}{|l|c|c|c|c|} \hline \text { County } & \begin{array}{c} \mathbf{2 0 0 0} \\ \text { population } \end{array} & \begin{array}{c} \mathbf{2 0 0 1} \\ \text { population } \end{array} & \begin{array}{c} \text { Change from 2000 } \\ \text { to 2001 } \end{array} & \begin{array}{c} \text { Percent } \\ \text { growth } \end{array} \\ \hline \text { Douglas County, CO } & 175,766 & 199,753 & & \\ \hline \text { Loudoun County, VA } & 169,599 & 190,903 & & \\ \hline \text { Forsyth County, GA } & 98,407 & 110,296 & & \\ \hline \text { Rockwall County, TX } & 43,080 & 47,983 & & \\ \hline \text { Wrilamson County, TX } & 249,967 & 278,067 & & \\ \hline \end{array}\) a. For each county, calculate the change in population from 2000 to 2001 , and use it to calculate the percent of growth. Which county had the largest percent of growth in this time period? (a) b. Los Angeles County in California is the largest county in the country. Its population was \(9,637,494\) in 2001 and \(9,519,338\) in 2000 . By what percent did the population of Los Angeles County grow? c. How does the growth of Los Angeles County compare to the population growth of the fastest-growing county? Which do you think is a better representation of the growth of a county, the percent of change or the actual number by which the county grew?

APPLICATION Last month it was estimated that a lake contained 3500 rainbow trout. Over a three-day period a park ranger caught, tagged, and released 100 fish. Then, after allowing two weeks for random mixing, she caught 100 more rainbow trout and found that 3 of them had tags. a. What is the probability of catching a tagged trout? b. What assumptions must you make to answer \(2 a\) ? c. Based on the number of tagged fish she caught two weeks later, what is the park ranger's experimental probability?

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)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.