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Let $E$ be the event that a randomly generated bit string of length $5$ starts with $0$, and let $F$ be the event that a randomly generated bit string of length $5$ has an odd number of $1’s$.

What is the conditional probability that a randomly generated bit string of length $5$ starts with $0$ given that the string has an odd number of $1’s$, i.e. the conditional probability of event $E$ given $F$? Are $E$ and $F$ independent events?

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closed as off-topic by Did, Mostafa Ayaz, Vladhagen, Delta-u, José Carlos Santos Sep 24 '18 at 17:53

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Assume that all the outcomes are equally probable; let the probability of each outcomes be $\frac 1{32}$.

Then $$\#E=\#\{00001,00010,00011,00100,00101,00110,00111,...,01111\}=16. $$

and

$$\#F=\#\{00001,00010,00100,00111,01000,01011,...,11111\}=16$$

and

$$\#E\cap F=\#\{00001,00010,00100,00111,01000,01011,01101,01110\}=8.$$

The conditional probability in question is

$$P(E\mid F)=\frac{P(E\cap F)}{P(F)}=$$ $$=\frac{P(\{00001,00010,00100,00111,01000,01011,01101,01110\})}{P(\{00001,00010,00100,00111,10000,01011,11001,11100,...,11111\})}=$$ $$=\frac{8/32}{16/32}=\frac12.$$


$E$ and $F$ are independent because $P(E\cap F)=\frac8{32}=\frac14$, $P(E)=\frac12$ and $P(F)=\frac{16}{32}=\frac12$ and $\frac12\times \frac12=\frac 14$

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  • $\begingroup$ @paw88789: Do you mean $P(E\mid F)=\frac 1{32}$? $\endgroup$ – zoli Sep 27 '15 at 11:40
  • $\begingroup$ The outcomes for $F \cap E$ are $01000, 00100,00010,00001,01110,01101,01011,00111$. Therefore $P(F \cap E)=\frac{8}{32}=\frac{1}{4}$ From where does 64 come from, since $2^5=32$ ? $\endgroup$ – callculus Sep 27 '15 at 11:55
  • $\begingroup$ @calculus: Sure, I have to edit quickly. There are $32$ possibilities as a total... THX. $\endgroup$ – zoli Sep 27 '15 at 12:00
  • $\begingroup$ @zoli Thanks, Can yo please check this question as well math.stackexchange.com/questions/1452840/… $\endgroup$ – Kittu Sep 27 '15 at 12:41
  • $\begingroup$ @Kittu: Please hit the check mark if you liked the answer. $\endgroup$ – zoli Sep 27 '15 at 12:42

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