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Chapter 4: Problem 63

Let \(A\) be the event that a number less than 3 is obtained if we roll a die once. What is the probability of \(A ?\) What is the complementary event of \(A\), and what is its probability?

Short Answer

Expert verified
The probability of event \(A\) is \(\frac{1}{3}\) and the probability of the complementary event \(A'\) is \(\frac{2}{3}\).

Step by step solution

01

Calculate Total Possible Outcomes

A die has six faces. Each roll of a die is independent and can result in any one of these six outcomes. Therefore, the total number of possible outcomes when rolling a die once is 6.
02

Identify Favorable Outcomes For Event \(A\)

Event \(A\), as defined in the exercise, is obtaining a number that's less than 3 when rolling a die. The outputs that satisfy this condition are 1 and 2. So, there are 2 favorable outcomes for event \(A\).
03

Calculate Probability of Event \(A\)

The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. Therefore, the probability of event \(A\) is \(\frac{2}{6} = \frac{1}{3}\).
04

Identify Complementary Event

The complementary event of \(A\), denoted by \(A'\), is the event that \(A\) does not occur. This means if we roll a die, we get a number that is not less than 3. The numbers that fulfill this condition are 3, 4, 5, and 6. Hence, there are 4 favorable outcomes for event \(A'\).
05

Calculate Probability of Complementary Event \(A'\)

The probability of event \(A'\) is calculated the same way as the probability of event \(A\). It's the ratio of the number of favorable outcomes for \(A'\) to the total number of possible outcomes. Hence, the probability of \(A'\) is \(\frac{4}{6} = \frac{2}{3}\).

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Key Concepts

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

Complementary Events
In probability, complementary events are a fascinating concept. When we talk about an event \(A\), its complement, \(A'\), is the scenario where \(A\) doesn't happen. It covers all other possible outcomes.
This way, if the probability of event \(A\) is known, the probability of \(A'\) is just what's left over.
  • The formula for the complementary event is: \( P(A') = 1 - P(A) \)
So, if rolling a die yields a number less than 3 (event \(A\)), the complementary event \(A'\) gives a number 3 or greater. This simple relationship helps solve many probability problems easily.
Favorable Outcomes
Understanding favorable outcomes is key in probability. When you're given an event, favorable outcomes are those specific results that lead to the event happening.
  • For event \(A\), these are the numbers less than 3: 1 and 2.
  • Thus, there are 2 favorable outcomes for \(A\).
A crucial concept is counting these outcomes correctly and determining their probability based on the total possible outcomes. This forms the foundation for calculating event likelihood.
Rolling a Die
Rolling a die is a classic example in probability, often used because of its simplicity and predictability. A standard die has 6 faces, numbered from 1 to 6, making it a great tool for learning.
Each roll of a die is a random, independent trial with six possible outcomes.
  • Each face has an equal probability of \(\frac{1}{6}\).
  • Rolling provides a clear method to visualize total and favorable outcomes.
This experiment also introduces concepts like fairness and independence, making it a staple in probability education.
Independent Events
Independent events in probability are outcomes where one event does not impact the other. Each die roll stands alone, not affecting any subsequent rolls.
For example, rolling a 2 now doesn鈥檛 influence what number shows up next.
  • Mathematically, two events \(A\) and \(B\) are independent if \( P(A \cap B) = P(A) P(B) \).
  • This means knowing one doesn't inform about the other.
Understanding this helps in more complex scenarios, where multiple events occur without affecting each other鈥檚 outcomes.

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Most popular questions from this chapter

A random sample of 2000 adults showed that 1320 of them have shopped at least once on the Internet. What is the (approximate) probability that a randomly selected adult has shopped on the Internet?

The probability that an open-heart operation is successful is .84. What is the probability that in two randomly selected open-heart operations at least one will be successful? Draw a tree diagram for this experiment.

A box contains 10 red marbles and 10 green marbles. a. Sampling at random from the box five times with replacement, you have drawn a red marble all five times. What is the probability of drawing a red marble the sixth time? b. Sampling at random from the box five times without replacement, you have drawn a red marble all five times. Without replacing any of the marbles, what is the probability of drawing a red marble the sixth time? c. You have tossed a fair coin five times and have obtained heads all five times. A friend argues that according to the law of averages, a tail is due to occur and, hence, the probability of obtaining a head on the sixth toss is less than \(.50 .\) Is he right? Is coin tossing mathematically equivalent to the procedure mentioned in part a or the procedure mentioned in part b? Explain.

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A man just bought 4 suits, 8 shirts, and 12 ties. All of these suits, shirts, and ties coordinate with each other. If he is to randomly select one suit, one shirt, and one tie to wear on a certain day, how many different outcomes (selections) are possible?
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