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# A coin is tossed twice. If the second throw results in a tail, a die is thrown. Describe the sample space for this experiment.

Last updated date: 16th Jul 2024
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Answer
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Hint: To solve this question we need to have a concept of probability to be more precise we need to know about the sample space which is possible outcomes. To solve the problem we will write the sample space of each throw or cases given to us, which are twice tossing of coin and once rolling of die. With the help of these two sample spaces we will be finding the sample space of individual acts and then will be merging the possible outcomes keeping the conditions in mind which are given in the problem.

Complete step by step answer:
The question asks us to find the sample space for the experiment in which firstly a coin is tossed twice and in which we need to consider the second throw which has resulted in a tail and then a dice is rolled. So on writing the sample space of the coin tossed twice we get:
Sample Space of the coin tossed twice=$\left\{ \text{HH, HT, TT, TH} \right\}$
As per the condition given in the question which says that, we need to consider the outcome in which the second toss is a Tail, the favourable outcome hence become:
Favourable outcome=$\left\{ \text{HT, TT} \right\}$
Now moving to the next throw which is rolling of the die. So the sample space for it is:
Sample Space of the die rolled is= $\left\{ 1,2,3,4,5,6 \right\}$
The favourable outcomes of the cases we got, so next step will be to merge the two favourable outcomes. On doing this we get:
Sample Space of the experiment is =$\left\{ \text{HT, TT, HT1, TT1, HT2, TT2, HT3, TT3, HT4, TT4, HT5, TT5, HT6, TT6 } \right\}$
$\therefore$ The sample space of the above experiment is $\left\{ \text{HT, TT, HT1, TT1, HT2, TT2, HT3, TT3, HT4, TT4, HT5, TT5, HT6, TT6 } \right\}$

Note: Whenever we write the sample space of an experiment we need to write all the possibilities which can be included, and then as per the condition we will be removing the outcomes from the set of sample space.