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A botanist has developed a new hybrid cotton plant that can withstand insects better than other cotton plants. However, there is some concern about the germination of seeds from the new plant. To estimate the probability that a seed from the new plant will germinate, a random sample of 3000 seeds was planted in warm, moist soil. Of these seeds, 2430 germinated. (a) Use relative frequencies to estimate the probability that a seed will germinate. What is your estimate? (b) Use relative frequencies to estimate the probability that a seed will not germinate. What is your estimate? (c) Either a seed germinates or it does not. What is the sample space in this problem? Do the probabilities assigned to the sample space add up to \(1 ?\) Should they add up to \(1 ?\) Explain. (d) Are the outcomes in the sample space of part (c) equally likely?

Step by step solution

01

Calculate Probability of Germination

First, we need to find the probability that a seed will germinate using relative frequencies. This is done by dividing the number of germinated seeds by the total number of seeds. Therefore, the probability is calculated as \( P(\text{germinate}) = \frac{2430}{3000} \). Simplifying this gives: \( \frac{2430}{3000} = 0.81 \). Hence, the probability that a seed will germinate is 0.81.

02

Calculate Probability of Not Germinating

To find the probability that a seed will not germinate, we subtract the probability of germination from 1. So, we calculate \( P(\text{not germinate}) = 1 - 0.81 = 0.19 \). Therefore, the probability that a seed will not germinate is 0.19.

03

Define and Assess Sample Space

The sample space in this problem contains two events: the seed germinates or the seed does not germinate. Mathematically, the sample space \( S = \{ \text{germinate}, \text{not germinate} \} \). Check if the probabilities of these events add up to 1: \( 0.81 + 0.19 = 1.0 \). Yes, they add up to 1 as expected because the probabilities of mutually exclusive events in a sample space should always total 1.

04

Determine if Outcomes are Equally Likely

The outcomes in the sample space are not equally likely because the probability of a seed germinating (0.81) is different from the probability of it not germinating (0.19). Thus, the events are not equally likely since their probabilities are not equal.

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Key Concepts

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

Relative Frequency

Relative frequency is a way to estimate probabilities based on past experiments, using the ratio of the number of times an event occurs to the total number of trials or observations.
In this case, relative frequency helps the botanist guess the likelihood that a cotton seed will germinate. Simply put, it allows us to make predictions based on observed data.

• Here, we count how many seeds germinated out of the total. This is 2430 out of 3000.
• This relative frequency gives us a probability estimate of 0.81.

This means 81% of the seeds germinated in this experiment. By observing how often an event happened in past trials, we can predict future occurrences. It's like saying, "If it happened often before, it's likely to happen again under similar conditions."
Using this approach helps in making informed decisions about the success rate of germination.

Sample Space

The concept of sample space is about defining all possible outcomes of a probability experiment.
In this example, the sample space consists of the two outcomes: a seed either germinates or it doesn't.
Therefore, we express it as follows:

• Germinate
• Not Germinate

It's crucial to make sure that the total probability of the sample space's outcomes equals 1, ensuring that it includes all possible situations. In this case, adding the probabilities of germination (0.81) and not germinating (0.19), indeed sums up to 1.
This confirms that our defined sample space is complete. Understanding sample spaces helps for systematically assessing all possible outcomes, ensuring that probability estimates are accurate and cover every scenario.

Mutually Exclusive Events

Mutually exclusive events are scenarios where the occurrence of one event means other events cannot happen at the same time.
In our germination problem, a seed cannot both germinate and not germinate simultaneously within the same instance.
These are distinct events, and when described as such:

• "Germinate" and "Not Germinate" are mutually exclusive.

The probabilities of mutually exclusive events should collectively sum up to 1. That means all the possibilities between them are accounted for, and the total likelihood can't exceed or be less than certainty, which is 1.
The understanding of mutually exclusive events ensures the correct addition of probabilities in experiments with distinct separate outcomes that cannot occur at the same time.

Equally Likely Outcomes

Equally likely outcomes refer to scenarios where every outcome in a sample space has the same probability of occurring.
When each possible result has an equal chance, the events are equally likely.
However, in the given seed germination experiment, the events are not equally likely.

• The probability of germination (0.81) is not equal to the probability of not germinating (0.19).

This tells us not all outcomes have the same chance of occurrence. Understanding whether outcomes are equally likely gives more insight into the nature of the events being studied, helping to identify any biases or tendencies within the examined process.