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What is the probability of rolling a sum of 7 with 3 die?

  • Last Updated : 21 Nov, 2021

Another word for probability is a possibility. It is a math of chance, that deals with the happening of a random event. The value is indicated from zero to one. In math, Probability has been introduced to predict how likely events are to occur. The meaning of probability is basically the scope to which something is to be expected to happen.

Probability

To understand probability more accurately, 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 possible outcomes is 1/6. As the possibility of happening any of an event is the same so there are equal chances of getting any likely number in this case it is either 1/6 or 50/3%.

  • Frequency interpretation: Probabilities are recognized as mathematically suitable estimations to long-run respective frequencies.
  • Subjective interpretation: A probability statement indicates the belief of some person regarding how certain an event is likely to occur.

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

There are different types of events based on different criteria. One of the types is an equally likely event and a complimentary event. Then there are impossible and sure events. One type is a simple and compound event. There are independent and dependent events, mutually exclusive, exhaustive events, etc. Let’s take a look at these events in detail.

  • Equally Likely Events: After rolling 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 obtaining any number in this case it is either 1/6 in fair dice rolling.
  • Complementary Events: There is a probability or possibility of only two outcomes which is an event will occur or not. Like a person will eat or not eat, buying a car or not buying a car, etc. are examples of complementary events.
  • Impossible and Sure Events: If the probability of happening of a likely event is 0, such an event is called an impossible event and if the probability of happening of a likely event is 1, it is called a sure event. In other words, the empty set ϕ is an impossible event and the sample space S is a sure event.
  • Simple Events: Any event consisting of a single point of the sample space is known as a simple event in probability. For example, if S = {56 , 78 , 96 , 54 , 89} and E = {78} then E is a simple event.
  • Compound Events: Opposite to the simple event, if any event contains more than one single point of the sample space then such an event is called a compound event. Considering the same example again, if S = {56, 78, 96, 54, 89}, E1 = {56, 54 }, E2 = {78, 56, 89} then, E1 and E2 represent two compound events.
  • Independent Events and Dependent Events: If the occurrence of any event is completely unaffected by the occurrence of any other event, such events are known as independent events in probability and the events which are affected by other events are known as dependent events.
  • Mutually Exclusive Events: If the occurrence of one event excludes the occurrence of another event, such events are mutually exclusive events i.e. two events don’t have any common point. For example, if S = {1, 2, 3, 4, 5, 6} and E1, E2 are two events such that E1 consists of numbers less than 3 and E2 consists of numbers greater than 4. So, E1 = {1, 2} and E2 = {5, 6} . Then, E1 and E2 are mutually exclusive.
  • Exhaustive Events: A set of events is called exhaustive, which means that one of them must occur.
  • Events Associated with “OR”: If two events E1 and E2 are associated with OR then it means that either E1 or E2 or both. The union symbol (∪) is used to represent OR in probability. Thus, the event E1 U E2 indicates E1 OR E2. If we have mutually exhaustive events E1, E2, E3 …En associated with sample space S then, E1 U E2 U E3 U … En = S
  • Events Associated with “AND”: If two events E1 and E2 are associated with AND then it means the intersection of elements that is common to both the events. The joining symbol (∩) is used to represent AND in probability. Thus, the event E1 ∩ E2 indicates E1 and E2.
  • Event E1 but not E2: It represents the difference between both events. Event E1 but not E2 show all the end results which are present in E1 but not in E2. Thus, the event E1 but not E2 is represented as E1, E2 = E1 – E2

What is the probability of rolling a sum of 7 with 3 die?

Solution:

When the three 6-faced regular dice are rolled, all the possible outcomes = 6 × 6 × 6 = 216.

To get the sum of the points shown on the three dice to be 7, these should be in the following ways,

(1, 1, 5 ) and this can be permuted in ( 3!/2!) = 3 ways, giving sum every time seven.

(1, 2 ,4 ) gives (3!)= 6 permutations, with sum 7.

(1, 3, 3) gives (3!/2!) = 3 cases with sum 7 and,

(2, 2, 3) gives similarly 3 cases. And these is all the favorable cases giving the sum seven and these are 

(3 + 6 + 3 + 3) = 15 cases .

Hence the required probability = 15/216 =5/72.

Similar Problems

Question 1: What is the probability of having a sum of 10 after rolling 3 dice?

Solution:

There are 63 possible outcomes to rolling a die 3 times. Out of these, how many yield a total of (exactly) 10 dots?

First find all sets {a, b, c} such that a + b + c = 10

  • 1, 3, 6
  • 1, 4, 5
  • 2, 3, 5
  • 2, 2, 6
  • 2, 4, 4
  • 3, 3, 4

The total number of sets that fit these criteria is 6. If a ≠ b ≠ c, then there exist 3! 

Unique permutations of {a, b, c}. If a = b ≠ c, then there exist 3 unique permutations of {a, b, c}: There cannot be a set such that a = b = c.

There are 3 sets of the first kind and 3 of the second. It follows that the total number of triple die rolls that can fit the criteria is

= 3 × 3! + 3 × 3

= 18 + 9

= 27

So, apply probability formula in it,

= {number of ways it can occur} ⁄ {Total number of outcomes}

= 27/216

= 1/8

Question 2: What is the probability of having a sum be at least 9 after rolling 3 dice?

Solution:

There are  6 × 6 × 6 = 216  ways to roll three dice. The ones that total 9 are (starting with the largest number) and ignoring the order for now:

  • 621
  • 531
  • 522
  • 441
  • 432
  • 333

Some of these can be achieved in multiple ways – put this number in brackets:

  • 621 (6)
  • 531 (6)
  • 522 (3)
  • 441 (3)
  • 432 (6)
  • 333 (1)

Adding up the numbers in brackets gives 25 ways of getting a total of 9. So the answer is: 25⁄216

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