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Chapter 5: Problem 13

Express the tossing of one coin as a probability distribution of \(x,\) the number of heads occurring (that is, \(x=1\) if a head occurs and \(x=0\) if a tail occurs).

Short Answer

Expert verified
The probability distribution of \(x\) is \(P(X=0) = 0.5\) and \(P(X=1) = 0.5\)

Step by step solution

01

Define the Discrete Random Variable

In this case, define \(x\) as the number of heads that occur in a single toss of a coin. Therefore, \(x\) can take either of the two possible values, 0 or 1. If a tail occurs \(x=0\), and if a head occurs \(x=1\).
02

Calculate the Probabilities

With a fair coin, the probability of each outcome (either heads or tails) is equal: \(P(X=0) = P(X=1) = 0.5\). This implies that the probability of getting a head ( \(X=1\) ) is 0.5 and the probability of getting a tail ( \(X=0\) ) is also 0.5.
03

Express as Probability Distribution

The probability distribution of \(x\) can now be expressed as follows: \(P(X=0) = 0.5\), \(P(X=1) = 0.5\). This represents the distribution of the probability of each outcome in a single toss of a coin.

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Key Concepts

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

Discrete Random Variable
In probability, a discrete random variable is a type of variable that can take on a countable number of distinct values. Consider the coin toss scenario, where the outcome is random and we're interested in the occurrence of heads. Here, the number of heads (\( x \) which can be either 0 or 1, is our discrete random variable. Unlike continuous variables, which can take on any value within a range, a discrete random variable has fixed possible values, much like the two faces of a coin.

When we articulate this in mathematical terms, we express \( x \) as a function that assigns specific values based on the outcome of the coin toss鈥�0 for tails and 1 for heads. These variables are the spine of discrete probability distributions, as they define the possible outcomes and their respective probabilities.
Probability of an Event
The probability of an event quantifies the likelihood of that event occurring, and in the setting of a coin toss, there are two mutually exclusive events鈥攆lipping heads or tails. Since a fair coin has no bias, the probability of landing on heads (\( x=1 \) is the same as that of landing on tails (\( x=0 \), which is 0.5 or 50%.

Understanding the concept of probability involves grasping that it is measured on a scale from 0 to 1, where 0 indicates impossibility and 1 signifies certainty. The probability helps us to predict how often we can expect an event to occur out of the total possible outcomes, providing a concrete way to express uncertainty and make informed predictions.
Binomial Probability
Now, let's step into the realm of binomial probability, a specific case of the probability distribution where there are exactly two possible outcomes (hence 'bi-nomial') each trial can produce. In the example of tossing a fair coin, the event of getting heads is what we term a 'success', while getting tails is a 'failure'. The binomial distribution requires the probability of success to be the same for each trial, just as with a coin toss, where each flip is independent of the others.

In broader terms, the binomial probability distribution is used when you are interested in the number of successes in a fixed number of repeated, independent trials. Since we have a fixed probability (\( p = 0.5 \) for getting heads on any given toss, this scenario fulfills all the criteria for binomial distribution. The formula to calculate the binomial probability would actually be overkill for our simple coin-flip example, but it becomes incredibly useful for more complex situations involving multiple trials.

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Most popular questions from this chapter

If \(x\) is a binomial random variable, calculate the probability of \(x\) for each case. a. \(\quad n=4, x=1, p=0.3\) b. \(\quad n=3, x=2, p=0.8\) c. \(\quad n=2, x=0, p=\frac{1}{4}\) d. \(\quad n=5, x=2, p=\frac{1}{3}\) e. \(\quad n=4, x=2, p=0.5\) f. \(\quad n=3, x=3, p=\frac{1}{6}\)

Let \(x\) be a random variable with the following probability distribution: $$\begin{array}{l|cccc}\hline x & 0 & 1 & 2 & 3 \\\P(x) & 0.4 & 0.3 & 0.2 & 0.1 \\\\\hline \end{array}$$ Does \(x\) have a binomial distribution? Justify your answer.

Find the mean and standard deviation for each of the following binomial random variables in parts a-c: a. The number of sixes seen in 50 rolls of a die b. The number of defective televisions in a shipment of 125 (The manufacturer claimed that \(98 \%\) of the sets were operative.) c. The number of operative televisions in a shipment of 125 (The manufacturer claimed that \(98 \%\) of the sets were operative.) d. How are parts b and c related? Explain.

a. Use a computer (or random number table) and generate a random sample of 100 observations drawn from the discrete probability population \(P(x)=\frac{5-x}{10},\) for \(x=1,2,3,4 .\) List the resulting sample. (Use the computer commands in Exercise \(5.36 ;\) just change the arguments. b. Form a relative frequency distribution of the random data. c. Form a probability distribution of the expected probability distribution. Compare the resulting data with your expectations. d. Construct a probability histogram of the given distribution and a relative frequency histogram of the observed data using class midpoints of \(1,2,3,\) and 4. e. Compare the observed data with the theoretical distribution. Describe your conclusions. f. Repeat parts a-d several times with \(n=100\) Describe the variability you observe between samples.

Did you ever buy an incandescent light bulb that failed (either burned out or did not work) the first time you turned the light switch on? When you put a new bulb into a light fixture, you expect it to light, and most of the time it does. Consider 8 -packs of 60 -watt bulbs and let \(x\) be the number of bulbs in a pack that "fail" the first time they are used. If 0.02 of all bulbs of this type fail on their first use and each 8 -pack is considered a random sample, a. List the probability distribution and draw the histogram of \(x.\) b. What is the probability that any one 8 -pack has no bulbs that fail on first use? c. What is the probability that any one 8 -pack has no more than one bulb that fails on first use? d. Find the mean and standard deviation of \(x .\) e. What proportion of the distribution is between \(\mu-\sigma\) and \(\mu+\sigma ?\) f. What proportion of the distribution is between \(\mu-2 \sigma\) and \(\mu+2 \sigma ?\) g. How does this information relate to the empirical rule and Chebyshev's theorem? Explain. h. Use a computer to simulate testing 1008 -packs of bulbs and observing \(x,\) the number of failures per 8 -pack. Describe how the information from the simulation compares with what was expected (answers to parts a-g describe the expected results). i. \(\quad\) Repeat part h several times. Describe how these results compare with those of parts a-g and with part h.
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