/*! 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 17 An athlete is shooting arrows at... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 17

An athlete is shooting arrows at a target. She has a record of hitting the centre \(30 \%\) of the time. Find the probability that she hits the centre a) with her second shot b) exactly once with her first three shots c) at least once with her first three shots.

Short Answer

Expert verified
a) 0.21, b) 0.441, c) 0.657.

Step by step solution

01

Define Probability for a Single Event

The probability that the athlete hits the center with a single shot is given as \( P = 0.3 \). Conversely, the probability of not hitting the center is \( 1 - P = 0.7 \).
02

Probability of First Two Shots (Part a)

For part a, we need the athlete to miss the first shot (probability \( 0.7 \)) and then hit the center on the second shot (probability \( 0.3 \)). Thus, the probability that she hits the center with her second shot, having missed the first, is obtained by multiplying these probabilities: \[ P(Miss \, 1st) \times P(Hit \, 2nd) = 0.7 \times 0.3 = 0.21. \]
03

Probability of Exactly One Hit in Three Shots (Part b)

For exactly one hit in three shots, we use the binomial probability formula \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\), where \(n=3\) shots, \(k=1\) hit, and \(p=0.3\). Substitute the values: \[ P(X=1) = \binom{3}{1} (0.3)^1 (0.7)^2 = 3 \times 0.3 \times 0.49 = 0.441. \]
04

Probability of At Least One Hit in Three Shots (Part c)

To find the probability of at least one hit, it is easier to calculate the probability of no hits in three shots and subtract from one. The probability of no hits (all misses) is calculated as \[ P(X=0) = \binom{3}{0} (0.3)^0 (0.7)^3 = 1 \times 1 \times 0.343 = 0.343. \]So, the probability of at least one hit is: \[ P( ext{at least one hit}) = 1 - P(X=0) = 1 - 0.343 = 0.657. \]

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.

Binomial Probability
Understanding binomial probability can be quite useful, especially when you're dealing with experiments that have two possible outcomes like success or failure. In the context of the shooting problem, the two outcomes are either hitting the target or missing it.
The binomial probability formula helps us calculate the likelihood of a given number of successes in a fixed number of attempts. This is especially useful when each attempt is independent of the others and has the same probability of success.
For our example, the probability of hitting the center with a single shot is 0.3 and missing is 0.7. To find the probability of the athlete hitting the center exactly once in three shots, we use the formula:
\[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}, \]
where:
  • \( n \) is the total number of shots (here 3),
  • \( k \) is the number of successful shots desired (here 1), and
  • \( p \) is the probability of success on a single shot (0.3).
This formula helps us understand how different outcomes can occur in fixed trials, repeatedly using the same process.
Shooting Problem
Shooting problems, like the one presented, are classic examples of applying probability to real-world situations. The athlete has a record probability (30%) of hitting the center.
Let's break it down by steps:
  • a) For her to hit the center on the second shot, she must miss on the first shot and hit on the second. This independent sequence has its own probability calculation: missing the first shot (0.7) multiplied by hitting the second (0.3), giving us 0.21 or 21% chance.
  • b) In scenarios where we want exactly one success in multiple trials, we apply binomial probability as mentioned. Here, calculating for 1 hit in 3 shots gave us a probability of 0.441.
  • c) The chance of hitting at least once in three shots is found by subtracting the probability of missing all shots from 1. This gives us 0.657, or a 65.7% probability at least one arrow will hit the center.
Understanding how to lay out and perform these calculations is vital for solving such problems.
Probability Calculation
Calculating probabilities involves careful analysis of given information and utilizing the right tools and formulas. First, recognize the type of problem; in binomial experiments like this, we have clear successes and failures.
1. **Individual shot probability**: Here, success is hitting the target (0.3 probability), and failure is missing (0.7 probability).
2. **Sequential outcomes**: Probability of hitting on specific attempts is found by considering the sequence of events like we did for part a). This requires multiplication of independent probabilities.
3. **Cumulative outcomes**: For calculating at least one success as in part c), smart use of complement probability (probability of at least one is 1 minus probability of none) simplifies the process.
Whenever implementing these principles, it's important to:
  • Define your trials and what constitutes success
  • Use appropriate formulas for specific scenarios
  • Double-check results by ensuring they fit logical expectations
These principles allow anyone to approach complex-sounding probability problems with confidence and precision.

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

When Olympic athletes are tested for illegal drug use (doping), the results of a single test are used to ban the athlete from competition. In an experiment on 1000 athletes, 100 were using the testosterone drug. During the medical examination, the available test would positively identify \(50 \%\) of the users. It would also falsely identify \(9 \%\) of the non-users as users. If an athlete tests positive, what is the probability that he/she is really doping?

In a large school, a student is selected at random. Give a reasonable sample space for answers to each of the following questions: a) Are you left-handed or right-handed? b) What is your height in centimetres? c) How many minutes did you study last night?

It is known that \(33 \%\) of people over the age of 50 around the world have some kind of arthritis. A test has been developed to detect arthritis in individuals. This test was given to a large group of individuals with confirmed cases and a positive test result was achieved in \(87 \%\) of the cases.That same test gave a positive test to \(4 \%\) of individuals that do not have arthritis. If this test is given to an individual at random and it tests positive, what is the probability that the individual has this disease?

Some young people do not like to wear glasses. A survey considered a large number of teenage students as to whether they needed glasses to correct their vision and whether they used the glasses when they needed to. Here are the results. $$\begin{array}{|l|c|c|c|} \hline \multirow{2}{*} & \multirow{2}{*}\multicolumn{2}{|c|}\text { Used glasses when needed }\text { Used glasses when needed } \\ \hline & & \text { Yes } & \text { No } \\ \hline \multirow{2}{*}\begin{array}{l} \text { Need glasses for } \\ \text { correct vision } \end{array} & \text { Yes } & 0.41 & 0.15 \\ \hline & \text { No } & 0.04 & 0.40 \\ \hline \end{array}$$ a) Find the probability that a randomly chosen young person from this group (i) is judged to need glasses (ii) needs to use glasses but does not use them. b) From those who are judged to need glasses, what is the probability that he she does not use them? c) Are the events of using and needing glasses independent?

An experiment involves rolling a pair of dice, 1 white and 1 red, and recording the numbers that come up. Find the probability a) that the sum is greater than 8 b) that a number greater than 4 appears on the white die c) that at most a total of 5 appears.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.