# What will be the maximum sum when two die are rolled simultaneously?

• Last Updated : 21 Dec, 2021

Probability is also known as a possibility. This means math of chance, that trade in the happening of a likely event. The value is designated from zero to one. In math, Probability has been shown to guess how likely events are to occur. Basically, the probability is the scope to which something is to be anticipated to happen.

### Probability

To understand probability more correctly, take an example as rolling a dice, the possible outcomes are – 1, 2, 3, 4, 5, and 6. The possibility of occurring any of the favorable outcomes is 1/6. As the probability of occurring any of the events is the same so there are similar chances of getting any likely number, in this case, it is either 1/6 or 50/3.

Formula of Probability

Probability of an event = {Number of favourable events } ⁄ {number of total events}

P(A) = {Number of ways A occurs} ⁄ {Total number of events}

DICE

Dice is a small block that has between one and six marks or dots on its edge and is used in games to give a random figure. Dice are small, tossable blocks with marked sides that can pause in several figures. They are used to give rise to random figures, often as part of sideboard games, as well as dice games, board games, role-playing games, and games of chance.

A customary die is a cube with each of its six faces noticeable with a different number of figures from one to six. When tossable or rolled, the die comes to pause appear a random number from one to six on its higher side, with the occurrence of each event being equally likely. Dice may also have concave or uneven shapes and may have faces marked with figures or characters instead of pips. Loaded dice are drawn to service some results over others for escape or entertainment.

### What will be the maximum sum when two die are rolled simultaneously?

Solution:

Possibility for throwing six sided two dice will be 1, 2, 3, 4, 5 and 6 faces in each (two) dice.

Two dice are roll at the same time, number of attainable outcome can be 62 = (6 × 6 ) = 36 because each die has 1 to 6 number on its dots.

Now, let’s consider the possible sums from rolling two dice. The smallest possible sum happens when all of the dice are the smallest, or one each. This gives a sum of two when two dice are thrown i.e., (1, 1). The greatest number on a die is six, which means that the greatest possible sum happens when all two dice are sixes is 12 i.e., (6, 6).

When n dice are rolled, the minimal attainable sum is n and the greatest attainable sum is 6n.

There is only one way when two dice can total 2,

• 2 ways for 3
• 3 ways for 4
• 4 ways for 5
• 5 ways for 6
• 6 ways for 7
• 5 ways for 8
• 4 ways for 9
• 3 ways for 10
• 2 ways for 11
• 1 ways for 12

Forming Sums,

As shown above, for two dice the possible sums include every number from 2 to 12.

Maximum total outcome of the two faces of the dice is (6, 6) = 6 + 6 = 12

The probability of obtaining maximum total outcome of the two faces of the dice = 1/36

### Similar Problems

Question 1: When two dice are rolled what is the probability of getting a total of 10 on the two faces of the dice?

Solution:

Possibility for throwing six sided two dice will be 1, 2, 3, 4, 5 and 6 faces in each (two) dies.

Two dice are roll at the same time, number of attainable outcome can be 6²= (6 × 6 ) = 36 because each die has 1 to 6 number on its dots.

Now, let’s consider the possible sums from rolling two dice. The smallest possible sum happens when all of the dice are the smallest, or one each. This gives a sum of two when two dice are thrown i.e., (1, 1). The greatest number on a die is six, which means that the greatest possible sum happens when all two dice are sixes is 12 i.e., (6, 6).

When n dice are rolled, the minimal attainable sum is n and the greatest attainable sum is 6n.

There is only one way when two dice can total 2,

• 2 ways for 3
• 3 ways for 4
• 4 ways for 5
• 5 ways for 6
• 6 ways for 7
• 5 ways for 8
• 4 ways for 9
• 3 ways for 10
• 2 ways for 11
• 1 ways for 12

Forming Sums,

As shown above, for two dice the possible sums include every number from 2 to 12.

The possibility of getting a total of 10 on the two face of the dice will be (5, 5) (6, 4) (4, 6)

Probability = 3/36

Question 2: When three dice are rolled the maximum total on the three faces of the dice will be?

Solution:

Possibility for throwing six sided three dice will be 1, 2, 3, 4, 5 and 6 dots in each (three) dies.

Three dice are roll at the same time, number of attainable outcome can be 63 = (6 × 6 × 6) = 216 because each die has 1 to 6 number on its dots.

Now, let’s consider the possible sums from rolling three dice. The smallest possible sum happens when all of the dice are the smallest, or one each. This gives a sum of three when three dice are thrown i.e., (1, 1, 1). The greatest number on a die is six, which means that the greatest possible sum happens when all three dice are sixes is 18 i.e., (6, 6, 6).

When n dice are rolled, the minimal attainable sum is n and the greatest attainable sum is 6n.

There is only one way when three dice can total 3,

• 3 ways for 4
• 6 ways for 5
• 10 ways for 6
• 15 ways for 7
• 21 ways for 8
• 25 ways for 9
• 27 ways for 10
• 27 ways for 11
• 25 ways for 12
• 21 ways for 13
• 15 ways for 14
• 10 ways for 15
• 6 ways for 16
• 3 ways for 17
• 1 ways for 18

Forming Sums,

As shown above, for three dice the possible sums include every number from three to 18.

Maximum total outcome of the three face die is (6, 6, 6) = 6 + 6 + 6 = 18

Probability = 1/216

Question 3: When two dice are rolled what is the possibility of getting a total of 5 on the two faces of the dice will be?

Solution:

Possibility for throwing six sided two dice will be 1, 2, 3, 4, 5 and 6 faces in each (two) dies.

Two dice are roll at the same time, number of attainable outcome can be 62 = (6 × 6 ) = 36 because each die has 1 to 6 number on its dots.

Now, let’s consider the possible sums from rolling two dice. The smallest possible sum happens when all of the dice are the smallest, or one each. This gives a sum of two when we are throwing two dice i.e., (1, 1). The greatest number on a die is six, which means that the greatest possible sum happens when all two dice are sixes is 12 i.e., (6, 6).

When n dice are rolled, the minimal attainable sum is n and the greatest attainable sum is 6n.

There is only one way when two dice can total 2,

• 2 ways for 3
• 3 ways for 4
• 4 ways for 5
• 5 ways for 6
• 6 ways for 7
• 5 ways for 8
• 4 ways for 9
• 3 ways for 10
• 2 ways for 11
• 1 ways for 12

Forming Sums,

As shown above, for two dice the possible sums include every number from 2 to 12.

The possibility of getting a total of 5 on the two face of the dice will be (4, 1), (1, 4), (2, 3), (3, 2).

Probability = 4/36

Question 4: When three dice are rolled what is the possibility of getting a total of 17 on the three faces of the dice will be?

Solution:

Possibility for throwing six sided three dice will be 1, 2, 3, 4, 5 and 6 dots in each (three) dies.

Three dice are roll at the same time, number of attainable outcome can be 63 = (6 × 6 × 6) = 216 because each die has 1 to 6 number on its dots.

Now, let’s consider the possible sums from rolling three dice. The smallest possible sum happens when all of the dice are the smallest, or one each. This gives a sum of three when three dice are thrown i.e., (1, 1, 1). The greatest number on a die is six, which means that the greatest possible sum happens when all three dice are sixes is 18 i.e., (6, 6, 6).

When n dice are rolled, the minimal attainable sum is n and the greatest attainable sum is 6n.

There is only one way when three dice can total 3,

• 3 ways for 4
• 6 ways for 5
• 10 ways for 6
• 15 ways for 7
• 21 ways for 8
• 25 ways for 9
• 27 ways for 10
• 27 ways for 11
• 25 ways for 12
• 21 ways for 13
• 15 ways for 14
• 10 ways for 15
• 6 ways for 16
• 3 ways for 17
• 1 ways for 18

Forming Sums,

As shown above, for three dice the possible sums include every number from three to 18

The possibility of getting a total of 17 on the three face of the dice will be (6, 6, 5) (6, 5, 6) (5, 6, 6)

Probability = 3/216

Question 5: When two dice are rolled what is the possibility of getting a total of 12 on the two faces of the dice?

Solution:

Possibility for throwing six sided two dice will be 1, 2, 3, 4, 5 and 6 faces in each (two) dies.

Two dice are roll at the same time, number of attainable outcome can be 62= (6 × 6 ) = 36 because each die has 1 to 6 number on its dots.

Now, let’s consider the possible sums from rolling two dice. The smallest possible sum happens when all of the dice are the smallest, or one each. This gives a sum of two when two dice are thrown i.e., (1, 1). The greatest number on a die is six, which means that the greatest possible sum happens when all two dice are sixes is 12 i.e., (6, 6).

When n dice are rolled, the minimal attainable sum is n and the greatest attainable sum is 6n.

There is only one way when two dice can total 2,

• 2 ways for 3
• 3 ways for 4
• 4 ways for 5
• 5 ways for 6
• 6 ways for 7
• 5 ways for 8
• 4 ways for 9
• 3 ways for 10
• 2 ways for 11
• 1 ways for 12

Forming Sums,

As shown above, for two dice the possible sums include every number from 2 to 12.

There is only one possibility to obtain a total of 12 on the two face of the dice will be (6, 6)

Probability = 1/36

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