Three dice are rolled together what is the probability of getting at least one 4

Probability for rolling three dice with the six sided dots such as 1, 2, 3, 4, 5 and 6 dots in each (three) dies.

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What is the probability of rolling at least one 4?
When 3 dice are rolled what is the probability of getting 4 each time?
What is the probability of getting a sum of at least 4?
What is the probability of rolling 3 dice?

When three dice are thrown simultaneously/randomly, thus number of event can be 63 = (6 × 6 × 6) = 216 because each die has 1 to 6 number on its faces.

Worked-out problems involving probability for rolling three dice:

1. Three dice are thrown together. Find the probability of:

(i) getting a total of 5

(ii) getting a total of atmost 5

(iii) getting a total of at least 5.

(iv) getting a total of 6.

(v) getting a total of atmost 6.

(vi) getting a total of at least 6.

Solution:

Three different dice are thrown at the same time.

Therefore, total number of possible outcomes will be 63 = (6 × 6 × 6) = 216.

(i) getting a total of 5:

Number of events of getting a total of 5 = 6

i.e. (1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1), (2, 1, 2) and (1, 2, 2)

Therefore, probability of getting a total of 5

Number of favorable outcomes
P(E1) = Total number of possible outcome

= 6/216
= 1/36

(ii) getting a total of atmost 5:

Number of events of getting a total of atmost 5 = 10

i.e. (1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1) and (1, 2, 2).

Therefore, probability of getting a total of atmost 5

Number of favorable outcomes
P(E2) = Total number of possible outcome

= 10/216
= 5/108

(iii) getting a total of at least 5:

Number of events of getting a total of less than 5 = 4

i.e. (1, 1, 1), (1, 1, 2), (1, 2, 1) and (2, 1, 1).

Therefore, probability of getting a total of less than 5

Number of favorable outcomes
P(E3) = Total number of possible outcome

= 4/216
= 1/54

Therefore, probability of getting a total of at least 5 = 1 - P(getting a total of less than 5)

= 1 - 1/54

= (54 - 1)/54

= 53/54

(iv) getting a total of 6:

Number of events of getting a total of 6 = 10

i.e. (1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2).

Therefore, probability of getting a total of 6

Number of favorable outcomes
P(E4) = Total number of possible outcome

= 10/216
= 5/108

(v) getting a total of atmost 6:

Number of events of getting a total of atmost 6 = 20

i.e. (1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1), (1, 2, 2), (1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2).

Therefore, probability of getting a total of atmost 6

Number of favorable outcomes
P(E5) = Total number of possible outcome

= 20/216
= 5/54

(vi) getting a total of at least 6:

Number of events of getting a total of less than 6 (event of getting a total of 3, 4 or 5) = 10

i.e. (1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1) (1, 1, 3), (1, 3, 1), (3, 1, 1), (1, 2, 2), (2, 1, 2), (2, 2, 1).

Therefore, probability of getting a total of less than 6

Number of favorable outcomes
P(E6) = Total number of possible outcome

= 10/216
= 5/108

Therefore, probability of getting a total of at least 6 = 1 - P(getting a total of less than 6)

= 1 - 5/108

= (108 - 5)/108

= 103/108

These examples will help us to solve different types of problems based on probability for rolling three dice.

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What is the probability of rolling at least one 4?

Originally Answered: What is the probability of getting at least one 4 when you throw 2 dices? Getting 4 in one is 1/6 and in second is again 1/6. Probability of getting 4 in both is 1/36.

When 3 dice are rolled what is the probability of getting 4 each time?

The probability of getting a 4 each time if a die is rolled 3 times is 1/216.

What is the probability of getting a sum of at least 4?

Answer: The probability of rolling two dice and getting a sum of 4 is 1/12. Let's find how likely we get a sum of 4 when we roll two dice simultaneously. So, when we roll two dice there are 6 × 6 = 36 possibilities. When we roll two dice, the possibility of getting number 4 is (1, 3), (2, 2), and (3, 1).

What is the probability of rolling 3 dice?

Probability for rolling three dice with the six sided dots such as 1, 2, 3, 4, 5 and 6 dots in each (three) dies. When three dice are thrown simultaneously/randomly, thus number of event can be 6³ = (6 × 6 × 6) = 216 because each die has 1 to 6 number on its faces.