Chapter 12.6, Problem 15HP

To determine

To calculate the sum that is most likely to occur when two dice are rolled and also make a probability distribution table. Then to make the probability graph.

Answer to Problem 15HP

The sum 7 is most likely to happen when 2 dice are rolled 12 times and the probability distribution table is,

X=sum of Dice	2	3	4	5	6	7	8	9	10	11	12
Probability	$\frac{1}{36}$	$\frac{1}{18}$	$\frac{1}{12}$	$\frac{1}{9}$	$\frac{5}{36}$	$\frac{1}{6}$	$\frac{5}{36}$	$\frac{1}{9}$	$\frac{1}{12}$	$\frac{1}{18}$	$\frac{1}{36}$

The probability graph is made from the data obtained from the probability distribution table for the confirmation of the answer.

Explanation of Solution

Given information:

Two dice are rolled twelve times.

Calculation:

At first, find the number of ways to get each sum by rolling both dice together

When the sum is 2 the possibility of getting number in dice is $\left(1,1\right)$ so number of ways is 1.

When the sum is 3 the possibility is $\left(1,2\right)\text{ and }\left(2,1\right)$ so number of ways 2.

When the sum is 4 the possibility is $\left(1,3\right),\left(3,1\right),\left(2,2\right)$ so number of ways is 3

When the sum is 5 the possibility is $\left(1,4\right),\left(4,1\right),\left(2,3\right),\left(3,2\right)$ so the number of ways is 4

Similarly, for the sum 6, 7, 8, 9, 10, 11 and 12 the number of ways are 5, 6, 5, 4, 3, 2 and 1.

The number of ways are obtained from the possibility assumption.

So, there are 36 possible sums from rolling of two dice

Now, the probability can be calculated when number of ways is divided by the possible sum

For example, when sum = 2 so probability = $\frac{1}{36}$

When sum = 3 so probability = $\frac{2}{36}$ = $\frac{1}{18}$

When sum = 4 so probability = $\frac{3}{36}=\frac{1}{12}$

By similar ways all the probability distribution can be obtained

Therefore, the probability distribution table for this is,

X=sum of Dice	2	3	4	5	6	7	8	9	10	11	12
Probability	$\frac{1}{36}$	$\frac{1}{18}$	$\frac{1}{12}$	$\frac{1}{9}$	$\frac{5}{36}$	$\frac{1}{6}$	$\frac{5}{36}$	$\frac{1}{9}$	$\frac{1}{12}$	$\frac{1}{18}$	$\frac{1}{36}$

The sum 7 is most likely to happen when two dice are rolled 12 times

Graph:

  

Interpretation:

Hence, the probability graph is made from the data obtained from the probability distribution table for the confirmation of the answer.