You roll a pair of six-sided dice and record the sum. (a)...

You roll a pair of six-sided dice and record the sum. (a) List all of the possible sums and determine the probability of rolling each sum. (b) Use technology to simulate rolling a pair of dice and record the sum 100 times. Make a tally of the 100 sums and use these results to list the probability of rolling each sum. (c) Compare the probabilities in part (a) with the probabilities in part (b). Explain any similarities or differences.

You roll a pair of six-sided dice and record the sum.
(a) List all of the possible sums and determine the probability of rolling each sum.
(b) Use technology to simulate rolling a pair of dice and record the sum 100 times. Make a tally of the 100 sums and use these results to list the probability of rolling each sum.
(c) Compare the probabilities in part (a) with the probabilities in part (b). Explain any similarities or differences.

Key Concepts

Sample Space

The sample space refers to the complete set of all possible outcomes when rolling a pair of six-sided dice. In this context, each outcome is a pair of numbers corresponding to the face-up sides on each die, and the sums of these numbers form the basis for determining probabilities.

Theoretical Probability Distribution

This concept involves calculating the probabilities of different sums occurring by analyzing the sample space. By counting the number of favorable outcomes for each sum and dividing by the total number of outcomes, one can create a theoretical probability distribution that shows exactly how likely each sum is to occur.

Experimental Probability

Experimental probability is obtained through simulation or actual trials, where the relative frequency of each outcome is recorded. In the context of rolling dice, this involves simulating the roll multiple times (e.g., 100 rolls) and using the tallied frequencies of sums to approximate the probabilities.

Law of Large Numbers

The Law of Large Numbers is a key idea in probability theory stating that as the number of trials or experiments increases, the experimental probability of an event tends to converge towards its theoretical probability. While simulation results may differ from theoretical calculations in small samples due to randomness, larger samples tend to provide a closer approximation.

Transcript

00:01 All right, so in this question, this is actually an activity -based question, but still the first one is theoretical, so let us just go over it.

00:08 We are going to roll a pair of six -sided dice and record the sum.

00:12 Now, what we have to do is we have to list all the possible sums and determine the probability of rolling each sum.

00:18 So what are the sums that are possible? the minimum is two, right? in both the dice, you can get at least one.

00:28 So the minimum sum is 1 plus 1 and the maximum is going to be 6 plus 6.

00:32 Right.

00:33 There is only one possible way to get 2 and only one possible way to get 12 also.

00:40 Only one possible way.

00:43 Okay, both of these will be 1.

00:46 And how about 3? you can get 3 either by 2 plus 1 or either by 1 plus 2.

00:52 Right.

00:54 So there are two possible ways.

00:55 So what is going to happen is this is going to keep on increasing.

00:58 One way for 2.

00:59 Two ways for three.

01:01 Similarly, you'll have three ways for four.

01:03 You can see over here.

01:04 It increases all the way till seven, right? how can you get seven? it is either one plus six or two plus five or three plus four or four plus three because there are two dyes...

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