What is the probability of getting 20 points with 6 dice?

Probability of getting 20 points with 6 dice, is 0.0904 or approximately 9%. Let's learn the probability formula first then get the solution for the above question.

Formula of Probability

The formula used to calculate the probability of the event is,

For example, if we flip a coin the probability of getting the head is 1/2 as the number of outcomes of the head is 1 and the total number of outcomes is 2.

What is Probability of Getting 20 Points with 6 Dice?

Solution:

Let probability of getting 20 with 6 dice be P.

To find the number of outcomes that result in a total score of 20, we can use a technique called generating functions. The generating function for a single die is (x + x² + x³ + x⁴ + x⁵ + x⁶), since each term represents the probability of rolling a specific value on the die. To find the generating function for 6 dice, we can simply multiply the generating function for a single die by itself 6 times:

(x + x² + x³ + x⁴ + x⁵ + x⁶)⁶

To find the sum 20 we have to find the coefficient of x²⁰ in (x + x² + x³ + x⁴ + x⁵ + x⁶)⁶

Take x common from the equation

Coefficient of x²⁰ in x⁶(1 + x + x² + x³ + x⁴ + x⁵)⁶

Coefficient of x¹⁴ in (1 + x + x² + x³ + x⁴ + x⁵)⁶

Using Sum of GP we get,

Coefficient of x¹⁴ in  [(1-x⁶) / (1-x)]⁶

Coefficient of x¹⁴  in (1 - x⁶)⁶  × (1 - x)^-6 ......(1)

Expansion of (1-x⁶)⁶ = 1 - (⁶C₁)×x⁶ + (⁶C₂)×x¹² - (⁶C₃)×x¹⁸+ .......

Coefficient of x¹⁴ so terms after x¹² will be useless because they are greater than x¹⁴ so we will ignore them.

Expansion of (1-x)^-6 = 1 + 6×x + (6×7×x²)/2! + (6×7×8×x³)/3! + .......

From eq(1)

Coefficient of x¹⁴ in [1 - (⁶C₁)×x⁶ + (⁶C₂)×x¹² - (⁶C₃)×x¹⁸+ .......]×[1 + 6×x + (6×7×x²)/2! + (6×7×8×x³)/3! + .......]

= ¹⁹C₁₄ - ⁶C₁×¹³C₈ + ⁶C₂×⁷C₂

= 4221

Therefore there are 4221 ways to get the sum of 20 in 6 dice

Total No. of ways = 6⁶ =46656

P = 4221/46656

   = 0.0904

Therefore the probability of getting the sum of 20 in 6 dice will be 0.0904

Read More,

• Probability Theory
• Probability Distribution

Similar Examples

Example 1: Find the probability of getting a sum of 8 when rolling two dice.

Solution:

Let the probability of getting  a sum of 8 is P(A)

Total no. of ways to get a sum of 8 when rolling two dice are,

• (2, 6)
• (3, 5)
• (4, 4)
• (5, 3)
• (6, 2)

Total number of possible outcomes when rolling two dice is 6 × 6 = 36, since each die has 6 possible outcomes.

Therefore, the probability P(A) = 5/36

Example 2: Find the probability of rolling at least one 6 when rolling two dice.

Solution:

Let the probability of getting at least one 6 in two dice be P(A)

Total no of ways to get at least one 6 when rolling two dice are,

• (1, 6)
• (2, 6)
• (3, 6)
• (4, 6)
• (5, 6) 
• (6, 1)
• (6, 2)
• (6, 3)
• (6, 4)
• (6, 5)
• (6, 6)

Total number of possible outcomes when rolling two dice is 6 × 6 = 36

Therefore, 

P(A) = Favourable / Total = 11/36

Example 3: If you flip a coin three times, what is the probability of getting exactly two tails?

Solution:

Let the probability of getting exactly two tails be P(A)

No of ways in which we can get three tails are:

• TTH
• THT
• HTT

Hence, there are total 3  ways to get three tails.

Total number of possible outcomes when flipping a coin three times is 2 × 2 × 2 = 8, since each flip has two possible outcomes.

Therefore,

P(A) = Favourable Case/ Total Case

P(A) = 3/8