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 :

Hint: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 $P(F) \ne 0$

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, $P\left( {\dfrac{E}{F}} \right) = \dfrac{{P(E \cap F)}}{{P(F)}}$.