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Chapter 14: Problem 41

Two fair dice are rolled. Determine the probability that the sum of the faces is \(7 .\)

Short Answer

Expert verified
The probability is \( \frac{1}{6} \).

Step by step solution

01

Identify Total Possible Outcomes

When two fair dice are rolled, each die has 6 faces. Thus, the total number of possible outcomes is calculated by multiplying the number of faces on each die: \[ 6 \times 6 = 36 \].
02

List the Successful Outcomes

Next, identify the outcomes where the sum of the faces equals 7. The possible pairs that sum to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Therefore, there are 6 successful outcomes.
03

Calculate the Probability

To find the probability, divide the number of successful outcomes by the total number of possible outcomes: \[ P(\text{Sum of 7}) = \frac{6}{36} = \frac{1}{6} \].

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Key Concepts

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

probability
Probability helps us understand how likely an event is to occur. It is a measure between 0 and 1.
If the probability is 0, the event will never happen. If it is 1, the event will always happen.
When calculating probability, we often use the formula: \( P(\text{Event}) = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}} \).
This formula works for any event, like rolling dice in our example.
dice outcomes
When rolling two fair dice, each with 6 faces, the possible outcomes include all the combinations of the faces.
There are 36 outcomes in total, calculated by multiplying the number of faces on each die, \(6 \times 6 = 36\).
Some examples of possible outcomes are (1,1), (2,3), and (6,6). Each combination is equally likely to occur in fair dice rolls.
sum of numbers
Now, let鈥檚 identify all outcomes that result in a sum of 7:
The pairs that give a sum of 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
These represent the successful outcomes for our event, leading to 6 successful outcomes in total.
The formula we learned earlier helps us find the probability of any sum.
fair dice
Fair dice are unbiased and have an equal probability of landing on any face.
Each face of the die shows up about 1/6 of the time in a large number of rolls.
This fairness is crucial for calculating probabilities, as each outcome is equally likely.
When using fair dice, probabilities reflect true randomness of the roll, making calculations like the sum of 7 accurate and reliable.

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Most popular questions from this chapter

Through observation, it has been determined that the probability for a given number of people waiting in line at the \(" 5\) items or less" checkout register of a supermarket is as follows: $$\begin{array}{lccccc}\begin{array}{l}\text { Number } \\\\\text { waiting in line }\end{array} & 0 & 1 & 2 & 3 & 4 \text { or more } \\\\\hline \text { Probability } & 0.10 & 0.15 & 0.20 & 0.24 & 0.31\end{array}$$ Find the probability of: (a) At most 2 people in line (b) At least 2 people in line (c) At least 1 person in line

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Use the binomial theorem to expand: \((x+2 y)^{5}\)

A baseball team has 15 members. Four of the players are pitchers, and the remaining 11 members can play any position. How many different teams of 9 players can be formed?

List all the combinations of 5 objects \(a, b, c, d,\) and \(e\) taken 2 at a time. What is \(C(5,2) ?\)

A coin is weighted so that heads is four times as likely as tails to occur. What probability should be assigned to heads? to tails?
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