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Chapter 5: Problem 10

Probability For each of the values, state whether the number could be the probability of an event. Give a reason for your answers. a. \(99 \%\) b. \(0.9\) c. \(9.9\) d. \(0.0099\) e. \(-0.90\)

Short Answer

Expert verified
a. Yes, \(99 \%\) is between 0 and 100%. b. Yes, \(0.9\) is between 0 and 1. c. No, \(9.9\) is not between 0 and 1. d. Yes, \(0.0099\) is between 0 and 1. e. No, probabilities cannot be negative.

Step by step solution

01

Analyzing option a. (99%)

99% is the same as 0.99, and it falls between 0 and 1. Therefore, 99% could be the probability of an event.
02

Analyzing option b. (0.9)

0.9 falls between 0 and 1. Therefore, 0.9 could be the probability of an event.
03

Analyzing option c. (9.9)

9.9 is greater than 1, and therefore it cannot be the probability of an event.
04

Analyzing option d. (0.0099)

0.0099 is close to 0 but still between 0 and 1. Therefore, 0.0099 could be the probability of an event.
05

Analyzing option e. (-0.90)

A probability can never be negative. Therefore, -0.90 cannot be the probability of an event.

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Key Concepts

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

Event Probability
Probability is all about chance or likelihood. It measures how likely it is for an event to happen. The probability of an event, like flipping a coin and landing on heads, is expressed as a number between 0 and 1.
  • A probability of 0 means the event is impossible.
  • A probability of 1 means the event is certain.
For example, rolling a fair 6-sided die to get any number between 1 and 6 has an equal chance. If you want to roll a 3, the probability is 1 out of 6, or approximately 0.167.
Understanding event probability is key in predicting outcomes and making decisions based on the likelihood of different events happening.
Probability Range
The range of probability is always between 0 and 1, inclusive. This means any valid probability must be in this range, either as a fraction, decimal, or percentage from 0% up to 100%.
  • Probabilities in the range can be expressed differently, like 50% is also 0.5.
  • Values greater than 1 or less than 0 are not valid probabilities.
Let鈥檚 highlight some examples from the exercise:
  • 0.9 is okay because it's less than 1 and more than 0. So, it fits well.
  • 99%, which is 0.99 in decimal form, is also within the range.
However, numbers like 9.9 or 150% are outside the valid range, so they're not possible as probabilities.
Negative Probability Values
Negative probabilities are not possible in real-world probability scenarios. By definition, probability means calculating how likely something is to happen, and negative means less than zero, which doesn鈥檛 make sense for likelihood.
When considering probabilities:
  • Negative numbers indicate a problem. Perhaps there's an error in calculations or a misunderstanding.
  • All valid probabilities should be non-negative, or in other words, greater than or equal to zero.
In the exercise, we saw the example of -0.9. Since it's negative, it cannot represent the probability of any event. The concept ensures clarity, avoiding confusion in determining the chance of occurrences.

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Most popular questions from this chapter

A Gallup poll conducted in 2017 asked people, "Do you think marijuana use should be legal?" In response, \(75 \%\) of Democrats, \(51 \%\) of Republicans, and \(67 \%\) of Independents said Yes. Assume that anyone who did not answer Yes answered No. Suppose the number of Democrats polled was 400 , the number of Republicans polled was 300 , and the number of Independents polled was 200 . a. Complete the two-way table with counts (not percentages). The first entry is done for you. Suppose a person is randomly selected from this group. b. What is the probability that the person is a Democrat who said Yes? c. What is the probability that the person is a Republican who said No? d. What is the probability that the person said No, given that the person is a Republican? e. What is the probability that the person is a Republican given that the person said No? f. What is the probability that the person is a Democrat or a Republican?

a. On a true/false quiz in which you are guessing, what is the probability of guessing correctly on one question? b. What is the probability that a guess on one true/false question will be incorrect?

Online Shopping A 2016 Pew Research poll reported that \(80 \%\) of Americans shop online. Assume the percentage is accurate. a. If two Americans are randomly selected, what is the probability that both shop online? b. If the two Americans selected are a married couple, explain why they would not be considered independent with regard to online shopping.

For each of the values, state whether the number could be the probability of an event. Give a reason for your answers. a. \(0.26\) b. \(-0.26\) c. \(2.6\) d. \(2.6 \%\) e. 26

a. Use the line of random numbers below to simulate flipping a coin 20 times. Use the digits \(0,1,2,3,4\) to represent heads and the digits 5 , \(6,7,8,9\) to represent tails. $$ \begin{array}{llll} 11164 & 36318 & 75061 & 37674 \end{array} $$ b. Based on these 20 trials, what is the simulated probability of getting heads? How does this compare with the theoretical probability of getting heads? c. Suppose you repeated your simulation 1000 times and used the simulation to find the simulated probability of getting heads. How would the simulated probability compare with the theoretical probability of getting heads?
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