Question

An experiment is run as follows- the colors red, yellow, and blue are each flashed on a screen for a short period of time. A subject views the colors and is asked to choose the one he feels was flashed for the longest time. The experiment is repeated three times with the same subject. a. If all the colors were flashed for the same length of time, find the probability distribution for \(x\), the number of times that the subject chose the color red. Assume that his three choices are independent. b. Construct the probability histogram for the random variable \(x\).

Short answer

Answer: The most likely number of times the subject will choose the color red in three trials is 1, as it has the highest probability of $\frac{12}{27}$.

Step-by-step solution

Identify the binomial distribution

Since the subject chooses one of the colors independently in each trial, and we are interested in the number of times the color red is chosen in three trials, we can model this problem as a binomial distribution with parameters \(n=3\) and \(p=\frac{1}{3}\) (the probability of choosing red).

Find the probabilities for each possible value of \(x\)

We have three trials, so there are four possible values for \(x\): 0, 1, 2, or 3. We will use the binomial probability formula to find the probabilities for each value of \(x\). The binomial probability formula is: \(P(x) = {n \choose x} \cdot p^x \cdot (1-p)^{(n-x)}\) Where \(n\) is the number of trials, \(x\) is the number of successful trials (the subject chooses red), \(p\) is the probability of success (probability of choosing red), and \({n \choose x}\) is the number of combinations of \(n\) items taken \(x\) at a time. For \(x=0\): \(P(0) = {3 \choose 0} \cdot \left(\frac{1}{3}\right)^0 \cdot \left(\frac{2}{3}\right)^3 = 1 \cdot 1 \cdot \frac{8}{27} = \frac{8}{27}\) For \(x=1\): \(P(1) = {3 \choose 1} \cdot \left(\frac{1}{3}\right)^1 \cdot \left(\frac{2}{3}\right)^2 = 3 \cdot \frac{1}{3} \cdot \frac{4}{9} = \frac{12}{27}\) For \(x=2\): \(P(2) = {3 \choose 2} \cdot \left(\frac{1}{3}\right)^2 \cdot \left(\frac{2}{3}\right)^1 = 3 \cdot \frac{1}{9} \cdot \frac{2}{3} = \frac{6}{27}\) For \(x=3\): \(P(3) = {3 \choose 3} \cdot \left(\frac{1}{3}\right)^3 \cdot \left(\frac{2}{3}\right)^0 = 1 \cdot \frac{1}{27} \cdot 1 = \frac{1}{27}\)

Present the probability distribution

The probability distribution for the random variable \(x\) is: \(x\) | \(P(x)\) ----|------ 0 | \(\frac{8}{27}\) 1 | \(\frac{12}{27}\) 2 | \(\frac{6}{27}\) 3 | \(\frac{1}{27}\)

Construct the probability histogram

To construct the probability histogram, we will represent the random variable \(x\) on the horizontal axis and the probabilities \(P(x)\) on the vertical axis. We will draw bars for each value of \(x\) with heights corresponding to their probabilities. The probability histogram will look like this: ``` | | [] | [] [][] | [][] [][] [] |------------------------- 0 1 2 3 ``` On the horizontal axis, we have the values of \(x\) (number of times the subject chose the color red). On the vertical axis, we have the probabilities \(P(x)\). The histogram shows the probabilities of each possible value of \(x\).

Key concepts

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In our example, the subject is asked to choose a color flashed on a screen, with each of the trials being independent. This scenario fits into a binomial distribution, a specific type of probability distribution.

In a binomial distribution, the parameters include:

• \( n \): number of trials (in this case, 3)
• \( p \): probability of success (probability of choosing red, which is \( \frac{1}{3} \))

The outcomes possible for the number of times the subject picks red (represented by \( x \)) are 0, 1, 2, or 3. Using the binomial probability formula:\[P(x) = {n \choose x} \cdot p^x \cdot (1-p)^{(n-x)}\]The probabilities for each outcome of \( x \) were calculated and representatively form the distribution:
- \( x = 0 \) : \( \frac{8}{27} \)
- \( x = 1 \) : \( \frac{12}{27} \)
- \( x = 2 \) : \( \frac{6}{27} \)
- \( x = 3 \) : \( \frac{1}{27} \)

Indicator Random Variable

The concept of an indicator random variable is essential in probability theory. It is a simple yet powerful way to represent the occurrence of an event. In this experiment, the indicator random variable can be used to denote whether the subject chose red or not in a given trial.

Here’s how it works:

• If the subject selects the color red, the indicator variable takes the value 1.
• If the subject selects any other color, the variable takes the value 0.

By using indicator random variables for each of the three trials, we can sum them up to find the total number of times red is chosen. This way, the problem is translated into a mathematical format that can easily be solved using the binomial distribution formula.

Probability Histogram

A probability histogram is a graphical way to represent a probability distribution. It provides a visual insight into how probabilities are distributed over different outcomes. In this experiment, the random variable \( x \) representing the number of times red is chosen, ranges from 0 to 3.

In the histogram:

• The horizontal axis represents the possible values of the variable (0, 1, 2, 3).
• The vertical axis denotes the probability of each value \( P(x) \).

Each bar in the histogram corresponds to one of the possible outcomes, and the height of the bar is proportional to the probability of that outcome. The resulting visual helps in quickly assessing which outcomes are more likely, providing a clear understanding of the distribution at a glance.