Two identical six-sided dice are rolled. What is the probability that

When two identical six-sided dice are rolled, in order for the sum to be less than 5,
the various combinations are (1,1),(1,2),(2,2),(2,1),(1,3),(3,1) - 6 combinations

The total possibilities are 6*6 = 36

Probability(Atleast 5) = 1 - P(Sum less than 5) = \(1 - \frac{6}{36} = \frac{30}{36} = \frac{5}{6}\)

Hence, the probability that the sum of the dice will be atleast 5 is \(\frac{5}{6}\)(Option D)

Hello!

Let's start with some probability basics.

Probability = Number of desired outcomes/Total number of possible outcomes

Because of this, probability is always between 0 and 1

The total number of possible outcome in rolling dice is 36

That is because there are 6 possible outcomes in the first die, and six possible outcomes for the second die.

Therefore there are 6 x 6 = 36 total possible outcomes

If we were to ask what is the probability of getting a picture of a cat on both dice, the probability is 0 because there are 0 outcomes that will feature a cat, unfortunately, out of the 36 possible outcomes

\(\frac{0}{36}\) = 0

If we were to ask what is the probability of getting a number from a roll of dice, that probability is 1 because there are 36 different ways to get a number out of the 36 possible outcomes of rolling dice. That is to say, all of the possible outcomes will result in a number. The probability is 1 meaning it is 100% a sure thing

\(\frac{36}{36}\) = 1


This question asks what the probability is of the sum of the dice being 5 or higher

Now if we start to think of the possible combinations

1 2
2 3
4 2
4 1
5 1
6 3
3 5

We can see that many add up to 5 or larger, so it is more useful to see how many rolls add up to less than 5 and subtract that number from the total number of rolls

Let's see:

1 1
1 2
1 3
2 1
2 2
3 1

There are 6 different rolls that will add up to less than 5

This is out of 36 total possible rolls

36 - 6 = 30 of the possible rolls will add up to 5 or larger

\(\frac{30}{36 } \)= \(\frac{5}{6}\\
\)

The answer is (D)

Two dice : 6 * 6 = 36 outcome

Sum at least 5 = 1 - sum less than 5

Sum less than 5 :

=> (1,1) (1,2), (1,3) = 3 results
=> (2,1) , (2, 2) = 2 results
=> (3,1) = 1 results

Sum less than 5 probability: \(\frac{6}{36} = \frac{1}{6}\)

Sum at least 5 = \(1 - \frac{1}{6} = \frac{5}{6}\)

Answer D

Given that Two identical six-sided dice are rolled and we need to find What is the probability that the sum of the dice will be at least 5?

As we are rolling two dice => Number of cases = \(6^2\) = 36

Out of these 36 outcomes if we are able to find the number of outcomes in which sum is less than 5 and subtract that from 36 then we will get the number of outcomes where we have sum at least 5

Following are the cases in which sum will be less than 5
(1,1), (1,2), (1,3)
(2,1), (2,2)
(3,1)

=> 6 cases

=> 36 - 6 = 30 cases have sum at least 5

=> Probability that Sum of two dice will be at least 5 = \(\frac{30}{36}\) = \(\frac{5}{6}\)

So, Answer will be D
Hope it helps!

Watch the following video to learn How to Solve Dice Rolling Probability Problems