Question 1163654
.
A coin is tossed 3 times. define the events A and B as:
A = {num heads > num tails}
B = {1st toss is a head}
Find P(A|B)
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<pre>
Let's write H for head and T for tail.


Then any outcome of 3 tosses can be written as a word of the length 3 written in this 2-letter alphabet,
consisting of letters H an T.


In all, there are 8 possible outcomes.  They all are listed as such words in the list below


1)   H H H     (*)    (**)   (***)

2)   H H T     (*)    (**)   (***)

3)   H T H     (*)    (**)   (***)

4)   H T T     (*)

5)   T H H            (**)

6)   T H T

7)   T T H

8)   T T T 


Notice that these outcomes have equal probability of  {{{(1/2)^3}}} = {{{1/8}}}  each.

By symbol (*), I marked that events where the 1st toss is a head.
There are 4 such events,  nn 1), 2), 3) an 4).
It is the set of events B.


By symbol (**), I marked that events where number of heads > number of tails.
There are 4 such events,  nn 1), 2), 3) an 5).
It is the set of events A.


By the definition of the conditional probability,

    P(A | B)  = P (A &#8745; B) / P(B).


The intersection  (A &#8745; B)  consists of three events  nn 1), 2) and 3)  in the list;  they are marked by the symbol (***).


Now,  P (A &#8745; B) = {{{3/8}}};  P(B) = {{{4/8}}};  therefore,

      P(A | B)  =  {{{((3/8))/((4/8))}}} = {{{3/4}}}.


<U>ANSWER</U>.  P(A | B) = {{{3/4}}}.
</pre>

Solved, explained and completed.


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The conception of conditional probability is not a simple conception, &nbsp;and a student should work diligently
to understand it clearly. &nbsp;For it, &nbsp;it is normal to consult/to learn from several sources.


If you want to learn more about the conditional probability, look into the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Conditional-probability-problems.lesson>Conditional probability problems</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/More-conditional-probability-problems.lesson>Conditional probability problems REVISITED</A> 

in this site.


You will find there many other similar solved problems to learn from.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic &nbsp;"<U>Solved problems on Probability</U>" 
and &nbsp;"<U>Additional problems on Probability</U>". 



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.



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