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# What is the probability of rolling the same number on 3 dice?

• Last Updated : 12 Oct, 2021

Probability is a part of math which deals with the possibility of happening random events. It is to predict how likely the events will occur or the event will not occur. The probability of happening an event is between 0 and 1 only and can also be written in the form of a percentage or fraction. The probability of event B is often written as P(B). Here P indicates the possibility and B indicate the happening of an event. Similarly, the probability of any event is often written as P(). When the end result of an event is not confirmed we use the probabilities of certain consequences—how likely they occur or what are the chances of their occurring.

The unpredictability of ‘probably’ etc., can be calculated numerically by means of ‘probability’ in many cases.

Though probability started with a bet, it has been used carefully in the fields of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather Forecasting, etc.

To understanding probability we take an example as rolling a dice:

There are six possible outcomes— 1, 2, 3, 4, 5, and 6.

The probability of getting any of the numbers is 1/6. As the event is an equally likely event so there is same possibility of getting any number in this case it is either 1/6 or 50/3%.

Formula of Probability

Probability of an event = {Number of ways it can occur} ⁄ {Total number of outcomes}

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

### Types of Events

Equally Likely Events: Rolling a dice the probability of getting any of the numbers is 1/6. As the event is an equally likely event so there is same possibility of getting any number in this case it is either 1/6 in fair dice rolling.

Complementary Events: Possibility of only two outcomes which is an event will occur or not. Like a person will eat or not eat the pizza, buying a bullet or not buying a bullet, etc. are examples of complementary events.

### What is the probability of rolling the same number on 3 dice?

Solution:

The three dice are fair, that are not biased in any manner whatsoever. The dice has 6 faces with a number between 1 and 6 inclusive, on each face with no matching numbers and no vacant faces.

The three dice are rolled fairly without any cheating.

Each of the dice rolls is an Independent Event, that is the outcome from anyone dice roll has no impact whatsoever on the outcome of any other dice roll.

The chance that any one die matches another is 1 out of 6 (1/6)

So the probability works out this way:

One fair die comes up any number = 1 (100%)

Another fair die matches that number = 1/6

Another matches the other two = 1/6

The probability of all three happening is the product of the three probabilities:

1 × (1/6) × (1/6) = 1/36.

### Similar Questions

Question 1: A coin is tossed 1000 times with the following frequencies:                                                                           Head: 455, Tail: 545

Compute the probability for each event.

Solution:

Since the coin is tossed 1000 times, the total number of trials is 1000. Let us check

the events of getting a head and of getting a tail as E and F, respectively. Then, the

number of times E happens, i.e., the number of times a head come up, is 455.

So, the probability of E = {Number of heads} ⁄ {Total number of trials}

i.e., P(E) = 455⁄1000 = 0.455

Similarly, the probability of the event of getting a tail = Number of tails ⁄ Total number of trials

i.e., P(F) = 545⁄1000 = 0.545

Note that in the above solution, P(E) + P(F) = 0.455 + 0.545 = 1 and E and F are the only two possible outcomes of each trial.

Question 2: What is the probability of rolling three dice and none match?

Solution:

For first throw, anything is possible and permissible.

So probability in first throw =1

For second throw, it has to be different than the first, meaning there are only 5

acceptable outcomes

So probability in second throw = 5/6

Similarly, in third throw, there are four acceptable outcomes

Probability in third throw = 4/6

So, altogether, we can say = 1 × 5/6 × 4/6 = 5/9

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