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Chapter 2: Problem 59

Suppose that two fair dice are tossed. What is the probability that the sum equals 10 given that it exceeds \(8 ?\)

Short Answer

Expert verified
The probability that the sum equals 10 given that it exceeds 8 is \(\frac{3}{10}\)

Step by step solution

01

Calculate the total number of outcomes

There are a total of \(6 \times 6 = 36\) possible outcomes when two dice are tossed.
02

Determine Outcomes where Sum Exceeds 8

To get a sum greater than 8, these combinations are possible: (3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6). So there are 10 outcomes in which the sum of the faces of two dice exceeds 8.
03

Determine Outcomes where Sum Equals 10

Out of the combinations obtained above, to get a sum equal to 10 we have: (4,6), (5,5), (6,4). Therefore, there are 3 outcomes in which the sum equals 10.
04

Compute the Conditional Probability

The formula to calculate the conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)}\). Hence, the probability that the sum equals 10 given that it exceeds 8 is given by \(\frac{3}{10}\).

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Key Concepts

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

Probability of Dice Outcomes
When dealing with dice, we often want to determine the likelihood of certain outcomes. Each die has six faces, so when rolling two dice, there are a total of 36 possible results. These results form a collection of all
outcomes that can happen, each equally likely. This means each single event, like rolling a specific number, has an identical chance of occurring, which is an essential part of calculating probability.
  • Imagine we're interested in the probability of obtaining a sum greater than a certain number, say 8, when two dice are rolled. We must first identify which pairs of rolls will produce such a result.
  • For instance, the combinations such as (4, 6) or (6, 6) contribute to sums greater than 8. There are several combinations possible, showing how outcomes compound.
  • In our specific example, knowing that 10 outcomes give a sum greater than 8 is key information for further calculations such as conditional probability.
Combinatorial Probability
Combinatorial probability delves into the number of ways events can occur, focusing on combinations and selections.
It applies when you鈥檙e interested in arrangements or groupings, such as possible sums of dice. Using combinations,
you determine how many specific outcomes exist within broader scenarios.
  • Take, for example, calculating the probability of specific sums resulting from dice rolls. This involves counting different pairings of dice rolls and organizing them into combinations.
  • Of interest is when the sum exactly totals 10: combinations like (4, 6), (5, 5), and (6, 4) present the necessary outcomes.
    This focused approach reveals exact probabilities within broader possibilities.
  • Such calculations are crucial for finding conditional probabilities, where knowing the individual components enables defining dependent outcomes.
Mathematical Statistics
Mathematical statistics embraces concepts of probability within broader theories of statistical analysis.
Understanding conditional probability is a significant aspect here. It determines the likelihood of an event happening
given that another has occurred, crucial in real-world applications.
  • The conditional probability formula is pivotal: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). This step-by-step calculation unveils the probability of sums equal to 10, assuming higher than 8.
  • In simpler terms, from all cases where the sum is greater than 8, what fraction also results in the sum being exactly 10? The answer simplifies the broader probability into specific desired outcomes.
  • Understanding such calculations helps in grasping interconnected probabilities, a common necessity in both academic studies and real-world decision making.

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Most popular questions from this chapter

A coke hand in bridge is one where none of the thirteen cards is an ace or is higher than a 9 . What is the probability of being dealt such a hand?

A fair coin is tossed four times. What is the probability that the number of heads appearing on the first two tosses is equal to the number of heads appearing on the second two tosses?

Let \(A\) and \(B\) be two events such that \(P\left((A \cup B)^{C}\right)=\) \(0.6\) and \(P(A \cap B)=0.1\). Let \(E\) be the event that either \(A\) or \(B\) but not both will occur. Find \(P(E \mid A \cup B)\).

A fair coin is tossed. If a head turns up, a fair die is tossed; if a tail turns up, two fair dice are tossed. What is the probability that the face (or the sum of the faces) showing on the die (or the dice) is equal to 6 ?

When Zach wants to contact his girlfriend and he knows she is not at home, he is twice as likely to send her an e-mail as he is to leave a message on her phone. The probability that she responds to his e-mail within three hours is \(80 \%\); her chances of being similarly prompt in answering a phone message increase to \(90 \%\). Suppose she responded within two hours to the message he left this morning. What is the probability that Zach was communicating with her via e-mail?
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