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Chapter 10: Problem 5

State whether each pair of events is dependent or independent. a. Roll a die, then roll the same die again. b. Remove one card from the deck, then draw a second card. (a) c. Flip a coin, then flip a second coin.

Short Answer

Expert verified
a. Independent b. Dependent c. Independent

Step by step solution

01

Understanding Independence

Two events are independent if the outcome of one does not affect the outcome of the other. Conversely, they are dependent if one outcome affects the likelihood of the other.
02

Analyzing the Die Roll

Rolling a die and then rolling it again is a classic example of independent events because the outcome of the first roll does not influence the outcome of the second roll.
03

Examining the Card Draws

When a card is removed from a deck and then another card is drawn, the events are dependent. This is because removing a card changes the probability of drawing any subsequent card since there are fewer cards remaining.
04

Discussing the Coin Flips

Flipping a coin and then flipping another coin are examples of independent events. Each coin flip is unaffected by the result of the preceding flip.

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Key Concepts

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

Probability
Probability is the measure of how likely an event is to occur. It quantifies uncertainty and ranges from 0 to 1. A probability of 0 means an event cannot happen, while a probability of 1 means the event is certain to happen.

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
  • Probability of event A = \( \frac{\text{Number of favorable outcomes for A}}{\text{Total number of possible outcomes}} \)
In the context of dependent and independent events, understanding probability helps us determine how one event might influence another or stand alone as having its chances unaffected. For example, when calculating the probability of getting a specific number on a dice roll, each roll is independent with equal chances per outcome.
Event Independence
Event independence is a crucial concept in probability. Two events are independent if the outcome of one event doesn't change the likelihood of the other.

Independent events satisfy the following condition:
  • \( P(A \text{ and } B) = P(A) \times P(B) \)
This means the probability of both events occurring together equals the product of their individual probabilities.

In our examples, rolling a die and then rolling it again represents independent events, as the result of the first roll doesn't affect the second. Comparing this to drawing cards from a deck, where drawing one card alters the deck's composition, we see how independence can be violated, making the events dependent.
Sample Space
The sample space of a random experiment consists of all possible outcomes. Understanding the sample space is vital in calculating probabilities and determining event independence.

For a single die roll, the sample space is \( \{1, 2, 3, 4, 5, 6\} \). Each outcome is equally likely in this uniform distribution.
  • When rolling a die twice, the sample space increases, consisting of all 36 combinations, such as (1,1), (1,2), ..., (6,6).
  • For card draws, if the sample space initially is all 52 cards, drawing one card changes the sample space for the next draw to 51 cards, influencing probabilities.
Understanding the sample space helps in recognizing event dependency, as any change in the initial sample space after an event can hint at a dependency relationship.
Random Experiments
Random experiments are activities or trials that result in one outcome from a set of possible outcomes. These experiments are foundational in studying probability.

Key features of random experiments are:
  • They produce unpredictable results.
  • The set of all possible outcomes constitutes the sample space.
Examples include rolling dice, flipping coins, and drawing cards from a deck. In the exercise examples, each trial鈥攂e it the dice roll, coin flip, or card draw鈥攊s a random experiment in itself.

Recognizing and understanding these random experiments help in setting up probability problems and analyzing event independence, ensuring that comparisons, such as those between independent dice rolls and dependent card draws, are well-founded.

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Most popular questions from this chapter

In a random walk, you move according to rules with each move being determined by a random process. The simplest type of random walk is a one-dimensional walk where each move is either one step forward or one step backward on a number line. a. Start at 0 on the number line and flip a coin to determine your move. Heads means you take one step forward to the next integer, and tails means you take one step backward to the previous integer. What sequence of six tosses will land you on the number-line locations \(+1,+2,+1,+2,+3,+2\) ? (a) b. Explore a one-dimensional walk of 100 moves using a calculator routine that randomly generates \(+1\) or \(-1\). In list \(\mathrm{L} 1\), generate random numbers with 1 representing a step forward and \(-1\) representing a step backward. Describe what you need to do with list L1 to show your number-line location after every step. [r \([\) See Calculator Note 10A and 10B. 4] c. Describe the results of your simulation. Is this what you expected?

APPLICATION Last month it was estimated that a lake contained 3500 rainbow trout. Over a three-day period a park ranger caught, tagged, and released 100 fish. Then, after allowing two weeks for random mixing, she caught 100 more rainbow trout and found that 3 of them had tags. a. What is the probability of catching a tagged trout? b. What assumptions must you make to answer \(2 a\) ? c. Based on the number of tagged fish she caught two weeks later, what is the park ranger's experimental probability?

Dr. Lynn Rogers of the North American Bear Center does research on bear cub survival. He observed 35 litters in 1996 . The distribution of cubs is shown in this table. Bear Litter Study \begin{tabular}{|l|c|c|c|c|} \hline Number of cubs & 1 & 2 & 3 & 4 \\ \hline Number of litters & 2 & 8 & 22 & 3 \\ \hline \end{tabular} (The North Bearing News, July 1997) a. Describe a trial for this situation. Name one outcome. (A) b. Is each outcome equally likely? Explain. (a) c. Based on the given information, what is the probability that a litter will have exactly three cubs? (a)

At this point in the season, Jackson has made 35 out of his 50 free throw attempts, so he's been successful on \(70 \%\) of his free throws. He wants to improve his rate to \(80 \%\) as soon as possible. a. How many consecutive free throws must he make to reach this goal? (a) b. If his probability of making any one shot is \(70 \%\), what is the probability that he will perform the number of consecutive free throws you found in 12a? (a)

Copy and complete this table to determine the expected value of the spinner game shown. \begin{tabular}{|l|c|c|c|c|} \hline Outcome & \(\$ 2\) & \(\$ 5\) & \(\$ 10\) & \\ \hline Probability & \(\frac{1}{3}\) & \(\frac{1}{2}\) & \(\frac{1}{6}\) & Sum \\ \hline Product & & & & \\ \hline \end{tabular}
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