I need help on this multiple choices question 2 Yahtzee is a

I need help on this multiple choices question.

2. Yahtzee is a game that involves rolling five dice to make certain combinations. The dice can be rolled up to three times in a turn. A player can save any dice that are wanted to complete a combination and then re-roll the other dice. A “yahtzee” is a five-of-a-kind and holds the game\'s highest point value (Ex: 3,3,3,3,3). A “full house” is a three-of-a-kind plus a pair (Ex: 3,3,3,5,5)

a) Suppose a player goes into the final roll of their turn with the values 3,3,3 and rolls 2 dice. What is the probability that the player gets a “full house”? Consider “yahtzee” as not a “full house”.

      A. 0.1388

      B. 0.1666

      C. 0.2578

      D. 0.4125

      E. None of the above.

b) The probability of rolling a yahtzee during a single turn is 0.04603. A game of Yahtzee consists of 13 turns per player. Find the probability that a player gets one or more “yahtzees” during a game. (Hint: It might be easier to find the probability that a player gets no “yahtzees” during a game.)

      A. 0.8438

      B. 0.6024

      C. 0.2500

      D. 0.4581

      E. 0.0156

Thank you,

Solution

Yahtzee is a game that involves rolling five dice to make certain combinations.

The dice can be rolled up to three times in a turn.

A player can save any dice that are wanted to complete a combination and then re-roll the other dice.

A “yahtzee” is a five-of-a-kind and holds the game\'s highest point value (Ex: 3,3,3,3,3). A “full house” is a three-of-a-kind plus a pair (Ex: 3,3,3,5,5)

a) Suppose a player goes into the final roll of their turn with the values 3,3,3 and rolls 2 dice. What is the probability that the player gets a “full house”? Consider “yahtzee” as not a “full house”.

we need three of one kind of dice, followed by a pair of a different type of dice. We will split this problem into two parts:

Once we know the number to each of these, we can multiply them together to give us the total number of full houses that can be rolled.

We begin by looking at the number of different types of full houses that can be rolled. Any of the numbers 1, 2, 3, 4, 5 or 6 could be used for the three of a kind.

There are five remaining numbers for the pair. Thus there are 6 x 5 = 30 different types of full house combinations that can be rolled.

For example, we could have 5, 5, 5, 2, 2 as one type of full house. Another type of full house would be 4, 4, 4, 1, 1. Another yet would be 1, 1, 4, 4, 4, which is different than the preceding full house because the roles of the fours and ones have been switched.

Now we determine the different number of ways to roll a particular full house. For example, each of the following give us the same full house of three fours and two ones:

4, 4, 4, 1, 1

4, 1, 4, 1, 4

1, 1, 4, 4, 4

1, 4, 4, 4, 1

4, 1, 4, 4, 1

We see that there are at least five ways to roll a particular full house. Are there others? Even if we keep listing other possibilities, how do we know that we have found all of them?

The key to answering these questions is to realize that we are dealing with a counting problem, and to determine what type of counting problem we are working with. There are five positions, and three of these must be filled with a four. The order in which we place our fours does not matter as long as the exact positions are filled. Once the position of the fours has been determined, the placement of the ones is automatic. For these reasons we need to consider the combination of five positions taken three at a time.

(5 C 3) = 5! / 3! (5-3)! = 10

This means that there are 10 different ways to roll a given full house.

Putting all of this together, we have our number of full houses. There are 10 x 30 = 300 ways to obtain a full house in one roll.

Since there are 300 ways to roll a full house in a single roll and there are 7776 rolls of five dice possible,

probabiliy of full house = 300 / 7776 = 0.03858

Option E) is correct.