Question

Two dice are thrown at the same time and the product of numbers appearing on them is noted. Find the probability that the product is a prime number.

Answer 1

Probability is the Ratio of number of Favorable Outcomes to the Total number of Possible Outcomes.

When Two Unbiased Dice are thrown at the same time and the numbers appearing on the Dice are Multiplied, we can come to a conclusion that the Total number of Possible Product numbers which are noted are : 6 × 6 = 36

⇒ Total number of Possible Outcomes : 36

The Question is to find the Probability that the Product is a Prime Number.

It means the Favorable Outcomes are those Product Numbers which are Prime.

For Example : One of the Favorable Outcome is the First Dice getting 1 and Second Dice getting 2 such that their Product is 1 × 2 = 2 which is a Prime Number.

In this Way : We can List out the Favorable Outcomes, they are :

             First Dice                  Second Dice                  Product

                   1                                 2                                     2

                   1                                 3                                     3

                   1                                 5                                     5

                   2                                1                                      2

                   3                                1                                      3

                   5                                1                                      5

So , the Total Number of Favorable Outcomes are : 6

Probability that the Product is a Prime Number is : $\frac{6}{36} = \frac{1}{6}$

Answer 2

Here is your answer:

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You may presumably know that :

Probability = Number of favourable outcomes / Number of total possibilities

To find probability you need to calculate two things:

• The number of possibilities
• The number of total favourable outcomes

The number of possibilities

When two dice are thrown , the number of possibilities are 36.

The first dice may show 1 , 2 , 3 , 4 , 5 , 6

For 1 the second dice has 6 possibilities (1,1) (1,2)(1,3),(1,4)(1,5),(1,6).

For the second the same will follow:

The observations will be :

{ (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)

(2,1,(2,2),(2,3),(2,4),(2,5),(2,6)

,(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)

,(4,1)(4,2)(4,3),(4,4),(4,5),(4,6)

,(5,1),(5,2),(5,3) ,(5,4),(5,5),(5,6)

,(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

Hence there will be exactly 36 possibilities.

Number of favourable outcomes

The prime number has only two factors:

• the number
• and itself.

The product has to be a prime.

So (1,2) , (2,1) (1,3) ( 3,1 ) ( 1,5 ) ( 5,1)

Products : 2 , 2 , 3 , 3 , 5, 5 are primes.

Hence there are exactly 6 observations.

(1,2) , (2,1) (1,3) ( 3,1 ) ( 1,5 ) ( 5,1)

Probability calculation

As discussed earlier,

P( for getting product of prime number ) =

number of outcomes favourable / number of possibilities

==>  6 / 36

==> 1 / 6

Hence the probability of getting a product of prime when two dice are thrown is 1 / 6

Hope it helps you

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