Consider the following game, played with three standard...

Consider the following game, played with three standard six-sided dice. If the player ends with all three dice showing the same number, she wins. The player starts by rolling all three dice. After this first roll, the player can select any one, two, or all of the three dice and re-roll them. After this second roll, the player can again select any of the three dice and re-roll them one final time. For questions (a) - (d), assume that the player uses the following optimal strategy: if all three dice match, the player stops and wins; if two dice match, the player re-rolls the die that does not match; and if no dice match, the player re-rolls them all. (a) Find the probability that all three dice show the same number on the first roll. (b) Find the probability that exactly two of the three dice show the same number on the first roll. (c) Find the probability that the player wins, conditioned on exactly two of the three dice showing the same number on the first roll. (d) By considering all possible sequences of rolls, find the probability that the player wins the game.

Consider the following game, played with three standard six-sided dice. If the player ends with all three dice showing the same number, she wins. The player starts by rolling all three dice. After this first roll, the player can select any one, two, or all of the three dice and re-roll them. After this second roll, the player can again select any of the three dice and re-roll them one final time. For questions (a) - (d), assume that the player uses the following optimal strategy: if all three dice match, the player stops and wins; if two dice match, the player re-rolls the die that does not match; and if no dice match, the player re-rolls them all.
(a) Find the probability that all three dice show the same number on the first roll.
(b) Find the probability that exactly two of the three dice show the same number on the first roll.
(c) Find the probability that the player wins, conditioned on exactly two of the three dice showing the same number on the first roll.
(d) By considering all possible sequences of rolls, find the probability that the player wins the game.

Key Concepts

Sample Space Analysis

This is the process of defining and understanding the complete set of possible outcomes in an experiment or game. It is vital for accurately calculating probabilities by considering every potential result from rolling dice.

Optimal Strategy

This involves selecting the best possible action at each decision point to maximize the chance of success. In probability games, determining the optimal strategy requires evaluating the outcomes of various decisions over multiple turns.

Combinatorics

This concept involves counting the number of ways events can occur, which is essential for determining the probabilities of specific outcomes when rolling dice. It underpins the calculation of the total number of possible results and the counts of favorable outcomes.

Conditional Probability

This refers to calculating the probability of an event occurring given that another event has already occurred. It is used when the outcome of one stage of the game affects the probabilities in subsequent stages.

Sequential Decision Making

This concept addresses how decisions made at different stages of a process or game influence the overall outcome. It involves analyzing a series of events or rolls, where the strategy adapts based on previous outcomes.

Transcript

00:02 All right, so here we have all the possible results from rolling two dice.

00:09 And the first part of the question is asking for the probability of us rolling a certain sum.

00:14 So we're going to add up the two dice after we roll them.

00:17 The first one that we have is the probability of rolling a seven.

00:21 And if we look, we are looking for all the different ways for us to make seven.

00:26 And we've got these ones here.

00:28 16, 25, 34, 435, 4352, and 6 .1.

00:36 Okay, so there are six total ways for us to make seven by rolling two dice, and there are a total of 36 possible outcomes.

00:46 This is a 6x grid, so there are 36 possible outcomes rolling two dice.

00:51 And so we're going to put 6 here, and if we put that into our calculator, we get 0 .1 ,6.

00:58 Six, seven.

01:00 Okay.

01:01 Again, each one of these answers is going to have 36 on the bottom because there are 36 total outcomes from rolling two dice.

01:09 The next one that we have is probably of rolling an 11, and there's only two ways here.

01:18 Okay, we've got 65 and 56.

01:21 So we're going to use those two.

01:25 And again, plugging that in to our calculators, we'll get 0 .05.

01:31 Six.

01:33 If we look for the probability of rolling a two, we see we just have the one possibility.

01:39 If we roll snake eyes or two ones, and so we just have one way, and that's going to give us 0 .028.

01:52 The next one that we have is three, and we've got one, two, and two one.

01:58 So again, we're only going to have two possible outcomes here, or two outcomes that result in a sum of three.

02:05 And that is going to be 0 .056.

02:10 The probability of us rolling a 12, the only way we get that is if we roll a 6 and a 6.

02:16 There's no other way.

02:17 So this is going to be 1 over 36, which is 0 .028.

02:25 And then finally, we have the probability of rolling 8.

02:29 And we've got a few of those, 2 -6 -35, 4 -4, 53, and 6 -2.

02:36 And that gives us a total of five possible outcomes.

02:41 So we have 0 .139.

02:45 Oops.

02:46 And actually, we'll go back, delete, release those marks.

02:55 Okay, and we'll rewrite that here.

02:59 5 .1 .339.

03:01 And now this next one is interesting.

03:03 We're being asked, what's the probability of us ruling doubles? or in other words, what's the probability of us rolling such that all of them are, or both dice are the same number? and we have quite a few cases here.

03:15 We've got 1 -1 -2 -2 -3 -4 -5 -5 and 6 -6.

03:25 And that means that there are six possibilities for us to roll doubles.

03:30 And so we're going to go ahead and get rid of those.

03:36 And we have 6, which gives us a .167 probability.

03:49 Now we're going to move on to the next set of questions.

03:53 The first one is the probability of rolling a 7 or an 11.

03:57 Now, rolling a 7, rolling an 11, these are what are called mutually exclusive events...

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