# If you roll a dice six times, what is the probability of rolling a number six?

• Difficulty Level : Expert
• Last Updated : 20 Oct, 2021

Probability is a part of mathematics that deals with the possibility of happening of the events. It is to forecast that what are the possible chances that the events will occur or the event will not occur. The probability as a number lies between 0 and 1 only and can also be written in the form of a percentage or fraction. The probability of likely event B is often written as P(B). Here P shows the possibility and B show the happening of an event. Similarly, the probability of any event is often written as P(). When the end outcome of an event is not confirmed we use the probabilities of certain outcomes—how likely they occur or what are the chances of their occurring.

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

To understand probability more accurately we take an example as rolling a dice:

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

The probability of getting any of the outcomes is 1/6. As the possibility of happening of an event is an equally likely event so there are same chances 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: After rolling a dice the probability of getting any of the likely events is 1/6. As the event is an equally likely event so there is some possibility of getting any number in this case it is either 1/6 in fair dice rolling.
• Complementary Events: There is a possibility of only two outcomes which is an event will occur or not. Like a person will play or not play, buying a laptop or not buying a laptop, etc. are examples of complementary events.

### If you roll a dice six times, what is the probability of rolling a number six?

Solution:

First you should find the probability that you will NOT get a 6 in order to find the probability that you will get a 6 at least once any of those times. This is much easier.

According to binomial concept

Let’s say P = probability of getting a 6 on each throw = 1/6.

P’ = probability of NOT getting a 6 on each throw is 1-p = 5/6.

When you want to calculate the probability of multiple (unconventional) events happening, you must multiply their independent probabilities (not add them).

So, The probability of not getting a 6 n times = P’ to the nth power.

In this case (5/6)6 = 15,625 / 46,656 ~ 0.334

But the probability we get is of NOT getting a 6 even once. And there  are only two possibilities: either we will see it at least once, or never see a 6. So the probability of getting at least one 6 is 1 minus this or about 0.666.

Note: It turns out this probability is roughly the same for any similar problem where you have a 1/n chance for an event and you try n times. In the limit as n approaches infinity, the probability of NOT getting it is 1/e ~ 0.36788. It is interesting that even at n = 6, it is not that far off.

### Similar Questions

Question 1: If a die is thrown 5 times, what is the probability of getting 6 exactly 3 times?

Solution:

According to binomial concept

The probability of 6 on one roll = 1/6

The probability of 6 on 3 rolls = (1/6)3

The probability of not 6 on one roll = 5/6

The probability of not 6 on 2 rolls = (5/6)2

Ways of selecting 3 from 5 = 5×4/2 =10

So

The probability of exactly 6 on 5 rolls = 10 × (1/6)3 × (5/6)2 = 0.0321

Question 2: What is the probability of at least one 6 when you throw 4 dice at the same time?

Solution:

According to binomial concept

The easiest way to think of this is first to think, “what is the probability of getting no 6’s when you roll 4 die?

In order to roll no 6’s in 4 rolls, you need to know the probability of not rolling a 6 with one dice.

Each 6 sided dice has 5 options that aren’t a 6 (1–5), giving not rolling a 6 a 5/6 chance with one fair dice on one toss.

So the probability of rolling no 6’s with 4 fair die, is (5/6)4

Therefore, the probability of rolling at least one 6 in 4 roll, is 1-(5/6)4 = 51.775%.

Question 3: What is the probability of getting at least one 6 if a die is rolled 3 times?

Solution:

According to binomial concept

Probability of getting at least one six

= 1 – probability of no six in three roll

Each 6 sided dice has 5 options that aren’t a 6 (1–5),

Giving not rolling a 6, a 5/6 chance with one fair dice on one toss.

So the probability of rolling no 6’s with 3 fair die, is (5/6)3.

= 1 – (5/6)3 = 0.42 (approx)

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