Work each problem. Two fair dice are rolled. Find the probabilities in parts (a)-(d). (a) The su

Work each problem. Two fair dice are rolled. Find the...

Key Concepts

Odds Against

Odds against are a way of expressing how unlikely an event is to occur. They are calculated as the ratio of the number of outcomes in which the event does not occur to the number of outcomes in which it does occur. Understanding odds is beneficial for interpreting probabilities in terms of risk or likelihood.

Sample Space

In probability theory, the sample space is the set of all possible outcomes of an experiment. For dice problems, the sample space typically consists of all possible pairs of numbers that can occur when two dice are rolled, which is a fundamental concept when determining probabilities.

Counting Techniques

Counting techniques involve using methods such as listing, multiplication principles, or combinatorial analysis to count the number of favorable outcomes relative to the total number of outcomes. This is crucial in discrete probability problems like dice rolling where outcomes are finite and usually uniformly distributed.

Probability Calculation

Probability calculation is the process of determining the likelihood of an event by dividing the number of favorable outcomes by the total number of outcomes in the sample space. This concept is foundational in solving any problem that asks for the probability of specific events occurring.

Union of Events

The union of events refers to the occurrence of at least one of multiple events. When solving problems where the outcome can satisfy one or more conditions (for example, an event that is 'either A or B'), it is important to understand how to correctly count and combine the outcomes without double counting.

Transcript

00:05 If you're of age and you've ever been into a casino where they had the game craps, then this problem is right up your alley because basically what you see in that table there, it's not in your textbook, at least not in the homework section, but you may want to pause for a moment and write the table down because if you ever come across problems like this again, this is the entire premise of rolling two dice, of the game of crap.

00:33 Caps.

00:35 Okay.

00:37 We have two different dice.

00:38 They're thrown simultaneously, or they're thrown simultaneously, and each die has sides of one through six.

00:46 And the game is to just add the dots together, add the numbers showing together.

00:54 Like, for instance, if one die had a three on it, the first die had a three on it and the second die had a four on it, that's right here.

01:02 Okay.

01:03 That's a seven.

01:07 Okay.

01:08 So that's all we're going to do is we're going to go through each of these probabilities and just highlight what we see inside of there or highlight the matching condition.

01:21 Okay.

01:22 Like so.

01:26 Oops.

01:28 Let me fix that.

01:32 That one is with this one.

01:38 All right.

01:39 So there are 36 possible outcomes.

01:42 That's what that n equals 36 in and purple means.

01:47 Okay.

01:48 So first probability, find the probability that the suburbality that the sub.

01:52 Sum of the dots is at least 10.

01:56 Okay.

01:57 Now the key phrase here is at least 10.

02:00 At least 10 means 10 or greater.

02:04 Well, in this case, the biggest number you can get is a 12.

02:08 So at least 10 means 10, 11, or 12.

02:13 So let's highlight the tens and the 11th and the 12th.

02:24 Well, how many numbers is that? well, let's see.

02:28 One, two, three, four, five, six.

02:31 So we have a six out of 36 probability, which, of course, reduces to one over six.

02:42 So the theoretical probability model dictates then that out of every six rolls, you should expect one of them to be ten or greater.

02:55 All right, and the second one, it says, the sum of the dots is less than 10.

03:02 Well, that would be the rest of them, wouldn't it? the ones that are highlighted in green, those are 10 or greater.

03:13 So the ones highlighted in yellow, those are the ones less than 10.

03:20 So if 6 out of 36 or 1 out of 6 is your probability of rolling a number at least 10, then the probability of rolling two dice that add up to less than 10 would be 5 out of 6, which should make sense because, again, theoretically, according to question a, if there's a probability of 1 out of 6 that you're going to roll a number 10 or larger, then the probability of not rolling a 10 or larger would be the other 5 out of 6...

Please give Ace some feedback