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Expectation of a Binomial Experiment: Suppose you conduct a random experiment that has a binomial probability distribution. Suppose the probability that outcome C occurs on any one repetition is \(0.4 .\) Let \(P(x)\) be the probability that outcome C occurs \(x\) times in five repetitions. a. Calculate \(P(x)\) for each value of \(x\) in the domain. b. Find the mathematically expected value of \(x\) (Hint: The value if \(C\) occurs \(x\) times is \(x .\) ) c. Show that the mathematically expected value of \(x\) is equal to 0.4 (the probability \(C\) occurs on one repetition) times 5 (the total number of repetitions). d. If the probability that \(C\) occurs on any one repetition is \(b\) and the probability that \(C\) does not occur on one repetition is \(a=1-b,\) prove that in five trials, the expected value of \(x\) is \(5 b\) e. From what you have observed in this problem, make a conjecture about the mathematically expected value of \(x\) in \(n\) repetitions, if the probability that C occurs on any one repetition is \(b\) f. If you plant 100 seeds, each of which has a probability of 0.71 of germinating, how many seeds would you expect to germinate?

Step by step solution

01

Calculate Binomial Probabilities

To calculate the binomial probabilities, use the formula for the probability of getting exactly k successes in n trials, which is given by: \( P(x) = \binom{n}{x} p^x (1-p)^{n-x} \), where \( p \) is the probability of success on an individual trial, \( n \) is the number of trials, and \( x \) is the number of successes. For this problem, \( n = 5 \) and \( p = 0.4 \). Calculate \( P(x) \) for \( x = 0, 1, 2, 3, 4, 5 \).

02

Compute the Expected Value

The expected value for a binomial distribution is given by \( E[X] = n \times p \). For this problem, use the previously calculated probabilities and compute the expected value of \( x \), which is the sum of each possible value of \( x \) times its corresponding probability, \( E[X] = \sum_{x=0}^{5} x \times P(x) \).

03

Demonstrate Expected Value using Probability

Show that the expected value of \( x \), calculated using probabilities in Step 2, is equivalent to 0.4 times 5. According to the formula \( E[X] = n \times p \), it should give \( E[X] = 5 \times 0.4 = 2 \). This should match with the sum calculated in Step 2.

04

General Expected Value with Variable Probability

To prove that for a probability \( b \) of success and with 5 trials, the expected value is \( 5b \), replace \( p \) with \( b \) in the expected value formula, yielding \( E[X] = 5 \times b \).

05

Conjecture for Expected Value in n Trials

From the steps before, the conjecture for the expected value of \( x \) in \( n \) trials with a probability \( b \) of success per trial is \( E[X] = n \times b \). This generalizes the calculation of expected value for any binomial distribution.

06

Calculate Expected Germination

If you plant 100 seeds and each has a probability of 0.71 of germinating, to calculate the expected number that will germinate, use the expected value formula with \( n = 100 \) and \( p = 0.71 \). Thus, the expected number of seeds to germinate is \( E[X] = 100 \times 0.71 = 71 \).

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Key Concepts

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

Binomial Experiment

A binomial experiment is a statistical experiment that satisfies the following four conditions: there must be a fixed number of trials, each trial must be independent of others, there are only two possible outcomes in each trial (often termed as 'success' and 'failure'), and the probability of success remains the same in each trial.

For example, tossing a coin five times is a binomial experiment. Here each toss is a trial with two possible outcomes (heads or tails), with a fixed probability of success (landing heads), performed with a fixed number of repetitions (five). Understanding this concept is crucial when calculating the probability distribution of random events in numerous fields, such as genetics, quality control, and survey sampling.

Expected Value

The expected value in statistics is the long-run average value of repetitions of the experiment it represents. It is a key concept in probabilistic analysis and provides a measure of the central tendency of a random variable.

The expected value, often denoted by E[X], for a binomial experiment can be computed as the product of the number of trials (n) and the probability of success on an individual trial (p). This concept is the foundation for predicting outcomes in probability theory and reinforces the role of probability in decision-making. For a binomial probability distribution with a probability of 0.4 for success across five trials, the expected value would be 5 multiplied by 0.4, equalling 2.

Binomial Probabilities

Binomial probabilities are the likelihoods of observing a specific number of successes in a binomial experiment. To calculate these probabilities, we use the binomial formula:
\[ P(x) = \binom{n}{x} p^x (1-p)^{n-x} \].

In this formula, \( \binom{n}{x} \) represents the number of ways to choose x successes from n trials (the binomial coefficient), p is the probability of success, and (1-p) is the probability of failure. By evaluating this formula for all possible values of x (from 0 to n), we create the binomial distribution, which is a discrete probability distribution that is very useful for predicting outcomes in scenarios with a binary outcome, such as our seed germination example.

Binomial Theorem

The binomial theorem is a fundamental mathematical theorem that describes the algebraic expansion of powers of a binomial. This theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \],

where \( \binom{n}{k} \) denotes the binomial coefficient. This theorem helps in finding the probability of obtaining a certain number of successes in a binomial experiment, and it can be applied to a wide range of problems, including those in algebra and probability theory. While it may seem abstract, the binomial theorem is the reason we can calculate such binomial probabilities so precisely. It provides a clear connection between binomial experiments and their associated probabilities and expected values.