Roll two dice and multiply the numbers. what is the probability that the product is a multiple of 6?

National University of Singapore

1. Jerry has a probability distribution for selling dishwashers. The probabilities are zero - 5%, one - 10%, two - 15%, three - 60%, four - 5%, and five - 5%. The mean of the distribution is a. 2.65 b. 2.75 c. 2.85 d. none of the above 2. Jim draws a card from a deck of cards and it is an ace. The probability that it is the ace of spades is a. 1/13 b. 4/52 c. 1/4 d, none of the above 3. Bill is trying to roll a three with a six sided die. The probability of Bill rolling the die three times and not rolling a three a single time is a. 1/216 b. 15/216 c. 125/216 d. none of the above

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.

What is the probability of rolling the product of 6 with 2 dice?

What is the probability of getting 6 in rolling a dice?

What are the multiples of 6?

What is the probability that the product will be 5 if two dices are thrown?