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Chapter 4: Problem 11

You roll two fair dice, a green one and a red one. (a) What is the probability of getting a sum of 6 ? (b) What is the probability of getting a sum of 4 ? (c) What is the probability of getting a sum of 6 or 4? Are these outcomes mutually exclusive?

Short Answer

Expert verified
(a) \(\frac{5}{36}\), (b) \(\frac{1}{12}\), (c) \(\frac{2}{9}\); Yes, mutually exclusive.

Step by step solution

01

Calculate Total Possible Outcomes

When two dice are rolled, each die has 6 faces. The number of total possible outcomes is calculated by multiplying the possible outcomes of each die: \(6 \times 6 = 36\).
02

Outcomes for Sum of 6

List the outcome pairs that result in a sum of 6: \((1,5), (2,4), (3,3), (4,2), (5,1)\). There are 5 such pairs.
03

Probability of Sum of 6

The probability is the number of favorable outcomes (sum of 6) over the total number of outcomes. \[ P(\text{sum of 6}) = \frac{5}{36} \]
04

Outcomes for Sum of 4

List the outcome pairs that result in a sum of 4: \((1,3), (2,2), (3,1)\). There are 3 such pairs.
05

Probability of Sum of 4

The probability is the number of favorable outcomes (sum of 4) over the total number of outcomes. \[ P(\text{sum of 4}) = \frac{3}{36} = \frac{1}{12} \]
06

Probability of Sum of 6 or 4

To find the probability of getting a sum of 6 or 4, we add the probabilities: \[ P(\text{sum of 6 or 4}) = P(\text{sum of 6}) + P(\text{sum of 4}) = \frac{5}{36} + \frac{1}{12} = \frac{5}{36} + \frac{3}{36} = \frac{8}{36} = \frac{2}{9} \]
07

Determine If Mutually Exclusive

The events of rolling a sum of 6 and rolling a sum of 4 are mutually exclusive because there are no outcomes that result in both a sum of 6 and 4 simultaneously with one roll of two dice.

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Key Concepts

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

Dice Outcomes
When you roll a pair of dice, each die has six faces, numbered from 1 to 6. This gives each die six different outcomes. In probability, these are called the possible outcomes. With two dice, a green one and a red one, we multiply the outcomes of each to find the total possible outcomes:
  • Green Die: 6 faces
  • Red Die: 6 faces
  • Total Outcomes: 6 x 6 = 36
These 36 outcomes represent every possible combination that can occur when both dice are rolled simultaneously. The outcomes range from (1,1), (1,2), ... up to (6,6). This foundational understanding helps in calculating the probabilities for more specific scenarios, such as sums of numbers.
Mutually Exclusive Events
In probability, certain events are mutually exclusive, meaning the occurrence of one event prevents the other from happening. In the context of rolling two dice, an important concept is whether two events can happen at the same time.
Let's examine the situations where the sum of the dice equals 6 and where the sum equals 4. Both cannot happen at the same time from a single roll of the dice:
  • Sum of 6: ( 1,5), (2,4), (3,3), (4,2), (5,1)
  • Sum of 4: (1,3), (2,2), (3,1)
There are no overlapping outcomes between these two sets. This confirms that the outcomes are mutually exclusive. Understanding this helps in calculating combined probabilities correctly without the need to subtract overlapping probabilities.
Favorable Outcomes
When we talk about favorable outcomes, we're referring to the outcomes that satisfy the condition we're interested in. For example, if you're looking to roll a sum of 6, you'll focus on those combinations that add up to 6:
  • Favorable Outcomes for Sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1)
Similarly, for a sum of 4:
  • Favorable Outcomes for Sum of 4: (1,3), (2,2), (3,1)
Counting these pairs gives us the number of favorable outcomes. This concept is key in determining the probability of an event as it represents the "successes" over the total possible occurrences.
Total Possible Outcomes
The total possible outcomes in a probability scenario involving dice are simply all the possible combinations that can occur. For two dice, this number is calculated by multiplying the number of outcomes for one die by the number of outcomes for the other die:
  • Each Die Outcomes: 6
  • Total Outcomes when rolling two dice: 6 x 6 = 36
This total number is vital because it forms the denominator when calculating the probabilities of specific events. If you have multiple possible outcomes, like different sums of dice, this total helps in understanding the likelihood of each specific result compared to the whole set of possibilities.

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Most popular questions from this chapter

The University of Montana ski team has five entrants in a men's downhill ski event. The coach would like the first, second, and third places to go to the team members. In how many ways can the five team entrants achieve first, second, and third places?

You toss a pair of dice. (a) Determine the number of possible pairs of outcomes. (Recall that there are six possible outcomes for each die.) (b) There are three even numbers on each die. How many outcomes are possible with even numbers appearing on each die? (c) Probability extension: What is the probability that both dice will show an even number?

If two events \(A\) and \(B\) are independent and you know that \(P(A)=0.3\), what is the value of \(P(A \mid B)\) ?

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