Question

Consider the events \(C_{1}, C_{2}, C_{3}\). (a) Suppose \(C_{1}, C_{2}, C_{3}\) are mutually exclusive events. If \(P\left(C_{i}\right)=p_{i}, i=1,2,3\), what is the restriction on the sum \(p_{1}+p_{2}+p_{3} ?\) (b) In the notation of part (a), if \(p_{1}=4 / 10, p_{2}=3 / 10\), and \(p_{3}=5 / 10\), are \(C_{1}, C_{2}, C_{3}\) mutually exclusive?

Short answer

The sum of the probabilities \( p_{1}+p_{2}+p_{3} \) for mutually exclusive events \(C_{1}, C_{2}, C_{3}\) must be less than or equal to 1. The given events \(C_{1}, C_{2}, C_{3}\) with probabilities \(p_{1} = 4/10, p_{2} = 3/10, p_{3} = 5/10\) are not mutually exclusive since their sum is greater than 1.

Step-by-step solution

Understand Mutually Exclusive Events

Mutually exclusive events in probability are events that cannot occur at the same time. In other words, if one event occurs, the other cannot. In this case, the events are \(C_{1}, C_{2}, C_{3}\). Since they are mutually exclusive, it means if \(C_{1}\) occurs, neither \(C_{2}\) nor \(C_{3}\) can occur, and so on.

Determine the Restriction on the Sum

The sum of the probabilities of mutually exclusive events gives the probability that any of these events will occur. However, the total probability cannot exceed 1. Therefore, the sum of the probabilities \(p_{1}+p_{2}+p_{3}\) should be less than or equal to 1. This is the restriction.

Check for Mutually Exclusive Events

Given that \(p_{1}=4 / 10, p_{2}=3 / 10\), and \(p_{3}=5/ 10\), the sum of these probabilities is \(4/10 + 3/10 + 5/10 = 1.2\). Since this sum is greater than 1, it indicates that the events \(C_{1}, C_{2}, C_{3}\) are not mutually exclusive as per the definition of mutually exclusive events in probability theory.

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random phenomena. The fundamental component of probability theory is an event, which refers to any particular occurrence or outcome that can take place when a random experiment is conducted. Events may be anything from rolling a specific number on a dice, drawing a certain card from a deck, or observing a particular weather condition.

Understanding events is crucial in developing the foundational knowledge required to calculate the likelihood of occurrences, known as probability. The probability of an event is a measure of the chance that the event will occur as a result of the experiment. It is quantified as a number between 0 and 1, inclusive, where 0 signifies an impossible occurrence, and 1 represents a certain outcome.

Event Probability

Event probability is a key concept that quantifies the likelihood of a specific event occurring within the framework of probability theory. When calculating the probability of an event, we are essentially asking, 'What is the chance that this event will happen?' The calculation often involves dividing the number of favorable outcomes by the total number of possibilities.

For instance, if we want to determine the probability of drawing an ace from a standard deck of cards, we divide the number of aces (4) by the total number of cards (52), giving us a probability of \( \frac{4}{52} \), or \( \frac{1}{13} \). To further clarify event probability, an important concept to understand is that of mutually exclusive events, which are events that cannot occur at the same time. For example, when flipping a coin, the events 'heads' and 'tails' are mutually exclusive because they cannot both occur on the same flip.

Probability Restrictions

Probability restrictions are rules or constraints that must be followed when calculating and interpreting probabilities. One of the fundamental restrictions in probability theory is that the sum of probabilities of all possible mutually exclusive outcomes of an experiment must equal 1. This is because one of those outcomes must occur, ensuring the complete set of outcomes is exhaustive.

When dealing with mutually exclusive events, such as in our example with the events \( C_{1}, C_{2}, C_{3} \), the sum of the probabilities \( p_{1} + p_{2} + p_{3} \) cannot be greater than 1. If it were, this would imply an overlap or that an additional outcome has been included, which would violate the very definition of mutually exclusive events. Such a violation suggests that either some of the events are not truly mutually exclusive, or there has been a miscalculation in the probabilities. This concept is often tested in educational exercises to ensure a thorough understanding of event interactions within sets of possible outcomes.