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Chapter 4: Problem 3

What is the probability of (a) an event \(A\) that is certain to occur? (b) an event \(B\) that is impossible?

Short Answer

Expert verified
(a) Probability of event A is 1. (b) Probability of event B is 0.

Step by step solution

01

Understanding Certain Events

An event that is certain to occur has a probability of 1, as it will happen every time. For example, when flipping a fair coin, the probability of either getting heads or tails is certain, hence the probability is 1.
02

Understanding Impossible Events

An event that is impossible has a probability of 0, as it will never happen. For instance, the probability of rolling a 7 with a standard six-sided die is impossible, so the probability is 0.
03

Applying Concepts to Event A

For event A that is certain to occur, the probability is given by the rule of certain events. Hence, the probability of event A is 1.
04

Applying Concepts to Event B

For event B that is impossible to occur, the probability is given by the rule of impossible events. Thus, the probability of event B is 0.

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Key Concepts

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

Certain Events
In probability, a certain event is an event that will definitely happen. Certain events are the outcomes that are guaranteed to occur, without any doubt.
For example, if you have a bag containing only red balls, the event of picking a red ball is certain since no other color is present.
The probability of a certain event is always 1 because it represents absolute certainty.
  • This means that if something has a certain probability, the chance of it not happening is zero.
  • In mathematical terms, a certain event's probability can be expressed as: \( P(E) = 1 \).
For instance, in a normal deck of cards, the event of drawing any card is a certain event, because you will definitely draw a card, and thus, \( P(E) = 1 \). Keeping this in mind helps you quickly identify situations of certainty in probability.
Impossible Events
An impossible event is one that cannot occur under any circumstances. This means the probability of an impossible event is zero.
Imagine, if you were to pick a card from a typical deck and asked about the probability of picking a card like an 'Ace of Stars' (which doesn't exist in a deck of cards), it would be impossible to draw such a card.
Thus, the probability of such an impossible event occurring is: \( P(E) = 0 \).
  • Impossible events highlight scenarios where there is absolutely no chance of an occurrence.
  • Always remember, there is no scenario in which an impossible event can occur.
Recognizing impossible events helps prevent mistakes in calculations and clarifies expectations, ensuring that unrealistic outcomes are not pursued.
Probability Rules
Probability rules are foundational to understanding and working with probability in any context. These rules guide how we calculate the likelihood of events. Among these, two primary rules directly apply to our discussion:
  • Certain Event Rule: As mentioned earlier, if an event is certain, the probability is \( P(E) = 1 \).
  • Impossible Event Rule: For events that cannot happen, the probability is \( P(E) = 0 \).
Beyond these two, probability rules include the rule of complement, and the addition rule for mutually exclusive events.
  • The complement rule helps calculate the probability of an event not occurring: \( P( ext{not } E) = 1 - P(E) \).
  • The addition rule explains how to compute probabilities for either of two mutually exclusive events occurring: \( P(A ext{ or } B) = P(A) + P(B) \).
Understanding these rules helps in accurately predicting outcomes and making informed decisions based on various scenarios in probability.

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Most popular questions from this chapter

What is the law of large numbers? If you were using the relative frequency of an event to estimate the probability of the event, would it be better to use 100 trials or 500 trials? Explain.

Compute \(C_{5,2}\).

You toss a pair of dice. (a) Determine the number of possible pairs of outcomes. (Recall that there are six possible outcomes for each die.) (b) There are three even numbers on each die. How many outcomes are possible with even numbers appearing on each die? (c) Probability extension: What is the probability that both dice will show an even number?

General: Candy Colors \(\mathrm{M} \& \mathrm{M}\) plain candies come in various colors. The distribution of colors for plain M\&M candies in a custom bag is $$\begin{array}{l|ccccccc} \hline \text { Color } & \text { Purple } & \text { Yellow } & \text { Red } & \text { Orange } & \text { Green } & \text { Blue } & \text { Brown } \\ \hline \text { Percentage } & 20 \% & 20 \% & 20 \% & 10 \% & 10 \% & 10 \% & 10 \% \\ \hline \end{array}$$ Suppose you have a large custom bag of plain M\&M candies and you choose one candy at random. Find (a) \(P\) (green candy or blue candy). Are these outcomes mutually exclusive? Why? (b) \(P\) (yellow candy or red candy). Are these outcomes mutually exclusive? Why? (c) \(P(\text {not purple candy}).\)

Consider a family with 3 children. Assume the probability that one child is a boy is 0.5 and the probability that one child is a girl is also \(0.5,\) and that the events "boy" and "girl" are independent. (a) List the equally likely events for the gender of the 3 children, from oldest to youngest. (b) What is the probability that all 3 children are male? Notice that the complement of the event "all three children are male" is "at least one of the children is female." Use this information to compute the probability that at least one child is female.
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