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,
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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
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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
