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What is the probability of getting a sum of 5 or 6 when a pair of dice is rolled?

  • Last Updated : 22 Sep, 2021

Probability is a numerical description of how likely an event is to occur. The probability of an event is in the range from 0 to 1 where 0 represents the impossibility of the event and 1 represents certainty over the thing. When the probability is higher, then there are more chances to occur the event. 

Terms used in Probability

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The terms used in probability are experiment, random experiment, sample space, outcome, and event. Let’s take a look at the  definitions of these terms in brief,



  • Experiment: An operation that produces some outcomes.

Example When we throw a die, there will be 6 numbers from which anyone can be up. So, the operation of rolling a die may be said to have 6 outcomes.

  • Random Experiment: An operation in which all possible outcomes are known but the exact outcome is not predictable.

Example When we throw a die there can be 6 outcomes but we cannot say the exact number which will show up.

  • Sample Space: All possible outcomes of an operation.

Example When we throw a die there can be six possible outcomes that is from {1,2,3,4,5,6} and represented by S.

  • Outcome: Any possible result out of the Sample Space S.

Example When we throw a die, we might get 6.

  • Event: Subset of a sample space that has to occur when an outcome belongs to an event and is represented by E.

Example When we roll a die there are six sample spaces {1, 2, 3, 4, 5, 6}. Let’s E occurs when “number is divisible by 2” then E ={2, 4, 6}. If the outcome is {2} which is a subset of E so it is considered an event that occurs otherwise event does not occur. Let’s look at the formula for an event occurring,

Probability of an event occur = Number of outcomes / Sample Space

What is the probability of getting a sum of 5 or 6 when a pair of dice is rolled?

Solution

Sample Space of one dice = 6



Sample Space of 2 dice = 6 × 6 = 36

Number of outcomes for sum of 5 = 4 {(1, 4), (2, 3), (3, 2), (4, 1)}

Number of outcomes for sum of 6 = 5 {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

Total Outcomes = 4 + 5 = 9

Probability of getting a sum of 5 or 6 = 9/36 = 1/4.

Sample Problems

Question 1: Probability of getting at least (minimum) one head while tossing two coins simultaneously.

Solution

Sample Space of one coin = 2

Sample Space of 2 coins = 2 × 2= 4

Number of outcomes for at least one head = 3 {(H, T),(T, H),(H, H)}

Probability of getting at least one head = 3/4.

Question 2: Probability of getting a sum of even number while rolling two dice.

Solution

Sample Space of one dice = 6

Sample Space of 2 dice = 6 × 6 = 36

Number of outcomes to get a sum of even = 18 ((1, 1),(1, 3),(1, 5),(2, 2),(2, 4),(2, 6),(3, 1),(3, 3),(3, 5),(4, 2),(4, 4),(4, 6),(5, 1),(5, 3),(5, 5),(6, 2),(6, 4),(6, 6))

Probability of getting a sum of even number = 18/36 = 1/2.

Question 3: Probability of getting a sum of multiple of 4 while rolling two dice.

Solution

Sample Space of one dice = 6



Sample Space of 2 dice = 6 × 6 = 36

Number of outcomes to get a sum of multiple of 4 = 9 ((1, 3),(2, 2),(2, 6),(3, 1),(3, 5),(4, 4),(5, 3),(6, 2),(6, 6))

Probability of getting a sum of multiple of 4 = 9/36 = 1/4.

Question 4: Probability of getting a product of 6 while rolling two dice.

Solution:

Sample Space of one dice = 6

Sample Space of 2 dice = 6 × 6 = 36

Number of outcomes to get a product of 6 = 4 ((1, 6),(2, 3),(3, 2),(6, 1))

Probability of getting a product of 6 = 4/36 = 1/9.

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