Question

die is thrown three times, and the sum of the three numbers thrown is 15. Find the chance that the first throw was a four.

Answer 1

Hint: In this question it is given that three dices are thrown together with the first dice being fixed with a number 4 on it so we will find the total number probability of getting a sum of 15 on the three dice by finding the ratio of the favorable outcomes by the total number of outcomes.

Complete step by step answer:
Given the sum of the numbers on the appearing faces of three dice is 15
Now it is said that the number appearing on the face of the first dice is 4, hence the sum of the numbers of the last two dice must be 15-4=11.
This means numbers appearing on the last two dice whose sum is 11 can be (6,5) or (5,6)
Hence the total number of favorable outcomes \[ = 2\]
Now we will find the total number of outcomes when the first dice is not fixed and the sum of the three faces are 15, so the numbers appearing on the dices can be \[\left\{ {\left( {4,5,6} \right),\left( {4,6,5} \right),\left( {5,4,6} \right),\left( {5,6,4} \right),\left( {6,4,5} \right),\left( {6,5,4} \right),\left( {6,3,6} \right),\left( {3,6,6} \right),\left( {6,6,3} \right),\left( {5,5,5} \right)} \right\}\]
So the total number of outcomes of getting sum of 15 on three dices \[ = 10\]
Hence the probability which is the ratio of total number of favorable outcomes and the total number of outcomes of an event will be
\[P\left( E \right) = \dfrac{2}{{10}} = \dfrac{1}{5}\]

Note: Probability means the certainty of occurring of any event. To find the probability of an experiment for which the outcomes can’t be guessed with certainty (Random experiment), two definitions are there, one is an event, and the other is sample space. Event is the other name of the favorable outcome of any experiment while Sample space is the set of all possible outcomes of that experiment and we can say that an event will be a subset of sample space.