There is a pair of rolled dice. what is the probability of 5 or 8 being the sum?

What is the probability of getting a sum of 5 or 8 when 2 dice are rolled once?

1. 1/3
2. 1/4
3. 5/35
4. 5/12

Answer (Detailed Solution Below)

Option 2 : 1/4

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Formula Used:

Probability = Possible outcomes/Total outcomes

Calculations:

When 2 dice are rolled, 

Total outcomes = 36

Possible outcomes of getting 5 = (1, 4), (4, 1), (2, 3), (3, 2)

⇒ Number of possible outcomes of getting 5 = 4

Possible outcomes of getting 8 = (2, 6), (6, 2), (3, 5), (5, 3), (4, 4)

⇒ Number of possible outcomes of getting 8 = 5

⇒ Total number of possible outcomes = 4 + 5 = 9

Probability = Possible outcomes/Total outcomes

⇒ Probability = 9/36 = 1/4

∴ The probability of getting a sum of 5 or 8 when 2 dice are rolled once is 1/4.

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So if you have 2 dice (supposing that they are 6 sided, we know that all the possible combinations of the dice are #36# So now we have to make a process of elimination:

#2+6# / #6+2# / #3+5# / #5+3# / #4+4# ( '/' seperates numbers)

We have repeated some because (let's say a dice is #a# and the other is #b#) #a# could be #3# but #b# could also be #3#.

So if we know that there are five know combinations for rolling sum of #8# then we write it as a fraction so #5/36#

We write it over the possible outcomes because of a simple rule:

the desired outcome / all possible outcomes

Hope this helped you out!

You can draw up a possibility space and count how many of the outcomes meet the requirements.

#{: (color(white)(0)," "ul1" "," "ul2" "," "ul3" "," "ul4" "," "ul5" "," "ul6" "),(1|," "2" "," "3" "," "4" "," "5" "," "6" "," "7" "),(2|," "3" "," "4" "," "5" "," "6" "," "7" "," "color(blue)(8)" "),(3|," "4" "," "5" "," "6" "," "7" "," "color(blue)(8)" "," "9" "),(4|," "5" "," "6" "," "7" "," "color(blue)(8)" "," "9" "," "10" "),(5|," "6" "," "7" "," "color(blue)(8)" "," "9" "," "10" "," "11" "),(6|," "7" "," "color(blue)(8)" "," "9" "," "10" "," "11" "," "12" ") :}#

There are #6xx6 =36# outcomes and of these there are #5# ways that a total of #color(blue)(8)# can be obtained.

#:. P(8) = 5/36#

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

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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