Question

Two different dice are rolled together. Find the probability of getting:(i) The sum of numbers on two dice to be 5.(ii) Even numbers on both dice(iii) A number greater than 3 on each die
(iv) A total of 6 or 7 of the numbers on two dice

Answer 1

We're given that two dice are rolled together at the same time.

Step 1: List out the possible outcomes.

$\begin{tabular}{| c | c | c | c | c | c | c |}\cline{1-7}& \sf 1 & \sf 2 & \sf 3 & \sf 4 & \sf 5 & \sf 6 \\\cline{1-7}\sf \ 1 \ & (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6)\\\sf 2 & (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6)\\\sf 3 & (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6)\\\sf 4 & (4,1) & (4,2) & (4,3) & (4,4) & (4,5) & (4,6)\\\sf 5 & (5,1) & (5,2) & (5,3) & (5,4) & (5,5) & (5,6)\\\sf 6 & (6,1) & (6,2) & (6,3) & (6,4) & (6,5) & (6,6)\\ \cline{1-7}\end{tabular}$

Therefore, the possible outcomes are;

(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)

∴ Total Number of outcomes: 36

-----------------------

Question 1: The sum of numbers on two dice to be 5.

Let's find the no: of favourable outcomes, i.e, the outcomes that will give 5 as the sum when the numbers on top of the dice are added.

(The outcomes in bold are the favourable outcomes)

(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)

∴ Favourable outcomes: (1,4) (2,3) (3,2) (4,1)

∴ No: of favourable outcomes: 4

Let "E" be the probability of this event.

$\sf \Longrightarrow Probability\Big(Event\Big) = \dfrac{No: of \ Favourable \ Outcomes}{Total \ Number \ of \ outcomes}$

$\sf \Longrightarrow Probability\Big(E\Big) = \dfrac{4}{36}$

$\sf \Longrightarrow Probability\Big(E\Big) = \dfrac{1}{9}$

-----------------------

Question 2: Even numbers on both dice.

Let's find the no: of favourable outcomes, i.e, the outcomes that have even numbers displayed on both the dice.

(The outcomes in bold are the favourable outcomes)

(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)

∴ Favourable outcomes: (2,2) (2,4) (2,6) (4,2) (4,4) (4,6) (6,2) (6,4) (6,6)

∴ No: of favourable outcomes: 9

Let "F" be the probability of this event.

$\sf \Longrightarrow Probability\Big(Event\Big) = \dfrac{No: of \ Favourable \ Outcomes}{Total \ Number \ of \ outcomes}$

$\sf \Longrightarrow Probability\Big(F\Big) = \dfrac{9}{36}$

$\sf \Longrightarrow Probability\Big(F\Big) = \dfrac{1}{4}$

-----------------------

Question 3: A number greater than 3 on each dice.

Let's find the no: of favourable outcomes, i.e, the outcomes that have both the numbers displayed greater than 3.

(The outcomes in bold are the favourable outcomes)

(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)

∴ Favourable outcomes: (4,4) (4,5) (4,6) (5,4) (5,5) (5,6) (6,4) (6,5) (6,6)

∴ No: of favourable outcomes: 9

Let "G" be the probability of this event.

$\sf \Longrightarrow Probability\Big(Event\Big) = \dfrac{No: of \ Favourable \ Outcomes}{Total \ Number \ of \ outcomes}$

$\sf \Longrightarrow Probability\Big(G\Big) = \dfrac{9}{36}$

$\sf \Longrightarrow Probability\Big(G\Big) = \dfrac{1}{4}$

-----------------------

Question 4: A total of 6 or 7 of the numbers on two dice.

Let's find the no: of favourable outcomes, i.e, the outcomes that give us a total of 6 or 7 when added.

(The outcomes in bold are the favourable outcomes)

(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)

∴ Favourable outcomes:

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

∴ No: of favourable outcomes: 11

Let "H" be the probability of this event.

$\sf \Longrightarrow Probability\Big(Event\Big) = \dfrac{No: of \ Favourable \ Outcomes}{Total \ Number \ of \ outcomes}$

$\sf \Longrightarrow Probability\Big(H\Big) = \dfrac{11}{36}$

-----------------------

Final answers:

• (i) 1/9

• (ii) 1/4

• (iii) 1/4

• (iv) 11/36