Question

Experiment II A sample space contains seven simple events: \(E_{1}, E_{2}, \ldots, E_{7} .\) Suppose that \(E_{1}, E_{2}, \ldots, E_{6}\) all have the same probability, but \(E_{7}\) is twice as likely as the others. Find the probabilities of the events. $$ B=\left\\{E_{1}, E_{3}, E_{5}, E_{7}\right\\} $$

Short answer

Answer: For \(E_1, E_2,..., E_6\), the probability for each event is \(\frac{1}{8}\), and for \(E_7\), the probability is \(\frac{1}{4}\).

Step-by-step solution

Assign variables

Let's assign the probability of \(E_1, E_2,..., E_6\) as \(p\). Then, the probability of \(E_{7}\), which has double the probability of the others, will be \(2p\).

Write the formula for the total probability

The total probability in the sample space should be equal to 1. So, the sum of the probabilities of \(E_1, E_2,..., E_7\) should be equal to 1. Let's write this equation: $$p + p + p + p + p + p + 2p = 1$$

Solve the equation

Combine the terms in the equation: $$6p + 2p = 1$$ $$8p = 1$$ Now, we can solve for \(p\): $$p= \frac{1}{8} $$

Calculate the probabilities of each event

Now that we know the value of \(p\), we can easily find the probabilities of all simple events by using the information provided. For \(E_1, E_2,..., E_6\), the probability for each event is the same as \(p\): $$P(E_1) = P(E_2) = \ldots= P(E_6) = \frac{1}{8} $$ For \(E_7\), the probability is double the value of \(p\): $$P(E_7) = 2p = 2 \times \frac{1}{8}$$ $$P(E_7) = \frac{1}{4} $$ Now we have the probabilities of all the events.

Key concepts

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events. In simple terms, it's the study of uncertainty and how to measure the likelihood of various outcomes.

For example, if we consider the flipping of a fair coin, there are two possible outcomes: heads or tails. Each outcome is equally likely, giving a probability of 0.5 or 50% for heads and the same for tails. Probability theory allows us to quantify these outcomes in a systematic way. It is foundational for many fields, including statistics, finance, gambling, science, and philosophy.

Simple Events Probability

In the realm of probability, simple events are the most basic possible outcomes that cannot be broken down into further outcomes. Each simple event is unique and represents an individual outcome. The probability of a simple event occurring is a numerical value between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty.

Considering our earlier sample space of seven simple events, each one, such as obtaining a specific face on a rolled die, is regarded as a simple event. The sum of the probabilities of all simple events in a sample space must equal 1, as they represent all possible outcomes of an experiment. Comprehending probabilities of simple events is crucial because complex events can often be expressed as combinations of these simple ones.

Total Probability Formula

The total probability formula is a key concept in understanding probability distributions in a finite sample space. It states that the total probability of all possible outcomes must add up to 1, reflecting the certainty that one of the outcomes will occur.

Our exercise example reflects this principle. With the probabilities of the first six events being equal and the last event having twice the probability, we theoretically distribute the total probability of 1 across all seven events proportionally. By setting up an equation as seen in the steps provided, we can resolve the value of each simple event's probability, ensuring the sum equals 1. This is essential, as the total probability formula serves as a validation tool for correctly calculating probabilities of events in a finite space.