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Chapter 4: Problem 2

Compute the probability of tossing a fair six-sided die and getting: a. an even number. b. a mumber less than 3 .

Short Answer

Expert verified
a. 1/2, b. 1/3

Step by step solution

01

Understand the Problem

We have a standard six-sided die numbered from 1 to 6. We need to calculate the probability of rolling an even number and then a number less than 3.
02

Calculate Probabilities for an Even Number

The even numbers on a die are 2, 4, and 6. There are 3 even numbers and the total number of possible outcomes when rolling a die is 6. The probability of rolling an even number is given by the formula \( P(\text{even}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \). Thus, \( P(\text{even}) = \frac{3}{6} = \frac{1}{2} \).
03

Calculate Probabilities for a Number Less Than 3

The numbers on a die that are less than 3 are 1 and 2, which gives us 2 favorable outcomes. Using the same probability formula, \( P(\text{less than 3}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \). So, \( P(\text{less than 3}) = \frac{2}{6} = \frac{1}{3} \).

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Key Concepts

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

Understanding a Six-Sided Die
A six-sided die, commonly known as a cubical die, is a popular tool used in games and probability exercises. Each face of the die is marked with a distinct number ranging from 1 to 6. This means that when you roll the die, there are exactly six possible outcomes.

  • Number of sides: 6
  • Possible outcomes: {1, 2, 3, 4, 5, 6}
  • Regular shape ensuring equal probability for each side

The fairness of the die indicates that each of these outcomes has an equal chance of occurring during a roll. Fairness is essential in probability calculations because it means that no side is weighted or favored over another. This attribute will help us calculate probabilities, such as rolling an even number or a number less than 3.
Discovering Even Numbers on a Die
An even number is any number that can be divided by 2 without leaving a remainder. On a standard six-sided die, the even numbers are 2, 4, and 6.

  • Even numbers on a die: 2, 4, 6
  • Total number of even outcomes: 3

The calculation of the probability of rolling an even number involves counting these possible outcomes. Since there are 3 even numbers and the die has 6 faces, we apply the probability formula: Number of favorable outcomes divided by the total number of outcomes. This results in: \[P(\text{even}) = \frac{3}{6} = \frac{1}{2}\] This means there is a 50% chance of rolling an even number on a six-sided die.
Identifying Favorable Outcomes
Favorable outcomes are the outcomes that satisfy the conditions of the event we are interested in. To compute probability, identifying the favorable outcomes is crucial. Let鈥檚 look at the example of rolling a number less than 3 on a six-sided die.

  • Numbers less than 3: 1, 2
  • Favorable outcomes: 2

To find the probability of rolling a number less than 3, we count the favorable outcomes, which are 1 and 2, giving us two successful results. Using our probability formula: Favorable outcomes over total outcomes, we get:\[P(\text{less than 3}) = \frac{2}{6} = \frac{1}{3}\] This indicates there is approximately a 33.3% chance of rolling a number less than 3 with a fair die. Identifying and counting favorable outcomes is a central process in understanding probability.

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Most popular questions from this chapter

What is the probability of rolling two fair six-sided dice a. and getting a sum greater than or equal to \(7 ?\) b. getting an even sum or a sum greater than \(7 ?\)

What is the probability of flipping a fair coin three times a. and getting a head each time? b. not getting a head at all?

At the local fair there is a game in which folks are betting where a chicken will poop on a 4 by 4 -foot grid. (There are 16,1 by 1 squares to choose from) You can buy a 1 by 1 -foot square for \(\$ 15\) and if the chicken poops on your square you win \(\$ 125\). Find the expected value for this game.

A professor gave a test to students in a science class and in a math class during the same week. The grades are summarized below. $$ \begin{array}{|c|c|c|c|c|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Total } \\ \hline \text { Science Class } & 7 & 18 & 13 & 38 \\ \hline \text { Math Class } & 10 & 8 & 9 & 27 \\ \hline \text { Total } & 17 & 26 & 22 & 65 \\ \hline \end{array} $$ If one student was chosen at random, find each probability: a. \(\mathrm{P}(\) in the math class) b. \(\mathrm{P}\) (earned a \(\mathrm{B}\) ) c. \(\mathrm{P}(\) earned an \(\mathrm{A}\) and was in the math class) d. \(\mathrm{P}(\) earned a \(\mathrm{B}\) given the student was in the science class) e. \(\mathrm{P}(\) is in the math class given that the student earned a \(\mathrm{B}\) )

Compute the probability of rolling five fair six-sided dice (each side has equal probability of landing face up on each roll) and getting: a. a 3 on all five dice. b. at least one of the die shows a 3.
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