Question

A coin tossed three times in succession. if $E$ is the event that there are at least two heads and$\;F$ is the event in which first throw is a head, then $P\left( {\dfrac{E}{F}} \right)$is equal to :

Answer 1

First we toss a coin, it will give 2 outcomes. They are Head or tail,
Here we tossed a coin 3 times so it will give $8$ outcomes. Now we find the possibilities of getting Event $E$ and Event $F$ and finally we find probability of $E$ and $F$.Here we use to find conditional probability, that is $P\left( {\dfrac{E}{F}} \right) = \dfrac{{P(E \cap F)}}{{P(F)}}$ , if $P(F) \ne 0$

Complete step-by-step answer:
It is given that A coin is tossed three times in succession.
$E$: event that there are at least $2$ heads.
$F$: event in which the first throw is head.
Total outcomes

1. H H H
2. H H T
3. H T H
4. T H H
5. T T H
6. T H T
7. H T T
8. T T T
   After tossing a coin three times we get $8$ outcomes
   $F$: event in which the first throw is head
   Total possibilities are $4$
   H H T or H T H or H T T or H H H
   The probability of the event F of getting head in the first trial is independent of the other $2$ trails so it becomes
   $P(F) = \dfrac{4}{8} = \dfrac{1}{2}$
   $E$: event that there are at least $2$ heads
   $2$ heads and $1$ tail or $3$ heads
   Now we find $P(E \cap F)$,
   The probability that the $3$ trails have at least $2$ heads when the first thrown is a head we can find it
   Total possibilities are $3$
   H H T or H T H or H H H
   $P(E \cap F) = \dfrac{3}{8}$
   $P(E/F) = \dfrac{{P(E \cap F)}}{{P(F)}}$
   $P(E/F) = \dfrac{{3/8}}{{1/2}} = \dfrac{3}{4}$
   Therefore, $P\left( {E/F} \right)$ is equal to $\dfrac{3}{4}$.

Note: If we toss a coin once the probability of getting a head or tail will be the same $= \dfrac{1}{2}$.When we toss a coin n times, we will get the total number of outcomes $= {2^n}$
Conditional Probability: If $E$ and $F$ are any two event in a sample space $S$ and $P(F) \ne 0$, then the conditional probability of given is, $P\left( {\dfrac{E}{F}} \right) = \dfrac{{P(E \cap F)}}{{P(F)}}$.