Two dice are rolled. Find the probabilities in parts...

Two dice are rolled. Find the probabilities in parts (a)-(d). (a). The sum of the dots is at least $10 .$ (b).The sum of the dots is less than 10 (c).The sum of the dots is either 7 or at least $10 .$ (d). The sum of the dots is $2,$ or the dice both show the same number. (e) .What are the odds against rolling a $7 ?$

Two dice are rolled. Find the probabilities in parts (a)-(d).
(a). The sum of the dots is at least $10 .$
(b).The sum of the dots is less than 10
(c).The sum of the dots is either 7 or at least $10 .$
(d). The sum of the dots is $2,$ or the dice both show the same number.
(e) .What are the odds against rolling a $7 ?$

Key Concepts

Odds versus Probability

Odds and probability are two different ways of expressing the likelihood of an event. While probability is defined as the ratio of the number of favorable outcomes to the total number of outcomes, the odds against an event are defined as the ratio of the number of unfavorable outcomes to the number of favorable outcomes. Understanding this relationship is crucial when interpreting questions that ask for odds instead of direct probabilities.

Compound Events and Union of Events

When an event is defined by multiple conditions, such as 'either one condition or another holds,' it is considered a compound event. Understanding how to combine events using the principles of probability, including the addition rule and the inclusion-exclusion principle, is important in situations where events may overlap, ensuring that all conditions are accurately accounted for in the overall probability.

Sample Space Enumeration

This concept involves listing all the possible outcomes in a probability experiment. In the context of rolling two dice, the sample space consists of all ordered pairs representing the result of each die roll. Understanding the sample space is essential because all probability calculations are based on the ratio of favorable outcomes to the total number of possible outcomes.

Counting Techniques

Counting techniques provide methods to determine the number of favorable outcomes without necessarily listing each possibility. Techniques such as the Fundamental Counting Principle, permutations, and combinations are used to efficiently count outcomes in experiments involving multiple random events, such as rolling dice, which then aids in calculating probabilities for various conditions.

Transcript

00:01 If i roll two dice, what is the probability that the sum of the dots is at least 10? well, to get 10, i could roll a 4 and a 6, a 5 in a 5, or a 6 and a 4.

00:15 Now, at least 10 means i could be bigger, so i could roll an 11.

00:19 And there's two ways i could do that, or i can roll a 12.

00:24 Six possibilities give me at least 10.

00:27 So the probability of rolling at least 10 is six options out of my total sample set.

00:38 Well, two dice each can go from one to six, 36 possibilities, which reduces to one -sixth.

00:45 So the probability of rolling at least a 10 is one -sixth.

00:50 What's the probability that i'll have less than a 10? well, that's going to be the complement.

00:58 The probability of an event and its complement always equals one.

01:02 So in this case, it's going to be 1 minus 1 6th or 5 6ths.

01:08 So the probability of at least 10 is 1 6th.

01:11 The probability of less than 10 is 5 .6th.

01:16 Next question.