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Chapter 12: Problem 5

Wardrobe In your dresser are five blue shirts, three red shirts, and two black shirts. a. What is the probability of randomly selecting a red shirt? b. What is the probability that a randomly selected shirt is not black?

Short Answer

Expert verified
The probability of drawing a red shirt randomly is 0.3 or 30%. The probability that a randomly selected shirt is not black is 0.8 or 80%.

Step by step solution

01

Identify the total number of shirts and red shirts

In the dresser, there are five blue shirts, three red shirts, and two black shirts. That's a total of \(5 + 3 + 2 = 10\) shirts. The red shirts are \(3\).
02

Calculate the probability of selecting a red shirt

The probability can be calculated by dividing the number of successful outcomes (red shirts) by the total number of outcomes (total shirts). So, the probability of selecting a red shirt is given by \(\frac{3}{10}\).
03

Calculate the probability of not selecting a black shirt

The total number of shirts that are not black is \(5 (blue) + 3 (red) = 8\). The total number of shirts is still \(10\). Thus, the probability of not selecting a black shirt is \(\frac{8}{10}\) or \(4/5\) when simplified.

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Key Concepts

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

Random Selection
Random selection is an essential concept in probability. It refers to choosing an item from a set where every item has an equal chance of being selected. Think of it like you have a collection of shirts in a drawer, and you reach in without looking to pick one. Each shirt has the same probability of being chosen.
In our example, when choosing a shirt from the dresser without looking, each shirt has an equal opportunity of being selected because the choice is made randomly. This ensures fairness in the selection process.
When we perform random selection, we are often checking the probability of picking specific items to understand how likely certain outcomes are.
Outcomes
In probability, an outcome is a possible result of a random selection process. When we consider probability scenarios like selecting a shirt from a drawer, each color shirt is an outcome of that selection.
For the exercise involving the dresser:
  • Outcomes for shirt colors can be blue, red, or black.
  • Total outcomes depend on the count of each type of shirt, which in this case adds up to 10 shirts in total.
Each different color represents a different potential outcome, and if we choose any shirt, any of these color outcomes could occur based on their relative frequency.
Simplification
Simplification in probability involves reducing fractions to their simplest form, making them easier to understand and interpret. This process does not change the value of the probability; it just presents it more clearly.
In our exercise, the probability of selecting a shirt that's not black was initially calculated as \(\frac{8}{10}\). By finding the greatest common divisor for the numbers 8 and 10, we can simplify \(\frac{8}{10}\) to \(\frac{4}{5}\).
Such simplification is helpful because it allows us to present results in a form that is more straightforwardly understood by everyone, especially when comparing different probabilities.
Successful Outcomes
In probability, a successful outcome is an event that satisfies the criteria of the event we're interested in. It is the specific result or results that we're hoping to achieve from our random selection.
For question (a) in our exercise, a successful outcome is selecting a red shirt. With three red shirts available, there are 3 successful outcomes. Therefore, the probability of a successful outcome, \(\frac{3}{10}\), results from dividing the number of successful outcomes (3 red shirts) by the total number of shirts (10).
This approach helps us understand the likelihood of achieving the desired result in a given scenario.

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Most popular questions from this chapter

Champion bowler A certain bowler can bowl a strike \(70 \%\) of the time. If the bowls are independent, what's the probability that she a. goes three consecutive frames without a strike? b. makes her first strike in the third frame? C. has at least one strike in the first three frames? d. bowls a perfect game (12 consecutive strikes)?

Car repairs A consumer organization estimates that over a 1-year period \(17 \%\) of cars will need to be repaired only once, \(7 \%\) will need repairs exactly twice, and \(4 \%\) will require three or more repairs. What is the probability that a car chosen at random will need a. no repairs? b. no more than one repair? c. some repairs?

Disjoint or independent? In Exercise 40 ?, you calculated probabilities involving various blood types. Some of your answers depended on the assumption that the outcomes described were disjint; that is, they could not both happen at the same time. Other answers depended on the assumption that the events were independent; that is, the occurrence of one of them doesn't affect the probability of the other. Do you understand the difference between disjoint and independent? a. If you examine one person, are the events that the person is Type \(\mathrm{A}\) and that the same person is Type \(\mathrm{B}\) disjoint, independent, or neither? b. If you examine two people, are the events that the first is Type A and the second Type B disjoint, independent, or neither? c. Can disjoint events ever be independent? Explain.

Lefties Although it's hard to be definitive in classifying people as right- or left-handed, some studies suggest that about \(14 \%\) of people are left- handed. Since \(0.14 \times 0.14=0.0196,\) the Multiplication Rule might suggest that there's about a \(2 \%\) chance that a brother and a sister are both lefties. What's wrong with that reasoning?

Winter Comment on the following quotation: "What I think is our best determination is it will be a colder than normal winter," said Pamela Naber Knox, a Wisconsin state climatologist. 鈥淧m basing that on a couple of different things. First, in looking at the past few winters, there has been a lack of really cold weather. Even though we are not supposed to use the law of averages, we are due." (Source: Excerpt from "The Law of Averages" by Naber Knox from Activity-Based Statistics: Student Guide by Richard L. Scheaffer, Jeffrey Witmer, Ann Watkins, Mrudulla Gnanadesikan. Published by Springer/Sci- Tech/Trade, ( \(\odot\) 1996.)
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