Question

A coin is tossed twice. Events E and F are defined as follows: E=heads on first loss, F=heads on second toss. Find the probability of $E \cup F$.
A) Hence $p\left( {E \cup F} \right) = \dfrac{1}{4}$
B) Hence $p\left( {E \cup F} \right) = \dfrac{3}{4}$
C) Hence $p\left( {E \cup F} \right) = \dfrac{1}{2}$
D) None of these

Answer 1

Probability means possibility. It is a branch of mathematics that deals with the occurrence of events. It is used to predict how likely events are to happen. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.
Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.
Random experiment is an experiment where we know the set of all possible outcomes but find it impossible to predict one at any particular execution.
Sample space is defined as the set of all possible outcomes of a random experiment. Example: Tossing a head, Sample Space(S) = {H, T}. The probability of all the events in a sample space adds up to 1.
Event is a subset of the sample space i.e. a set of outcomes of the random experiment.
${\text{Probability of an event}} = \dfrac{{{\text{Number of occurence of event A in S}}}}{{{\text{Total number of cases in S}}}}{\text{ }}$
$= \dfrac{{n\left( A \right)}}{{n\left( S \right)}}$

Complete step-by-step answer:
Let H denotes head and T denotes tail.
Sample space of tossing two coins S = {(H,H),(H,T),(T,H),(T,T)}
Total number of sample points n(S) = 4
Now, E = Head on first toss = {(H,H),(H,T)}
and F = Head on second toss = {(H,H),(T,H)}
\begin{gathered} E \cup F = \left\\{ {\left( {H,H} \right),\left( {H,T} \right),\left( {T,H} \right)} \right\\} \\\ \therefore n\left( {E \cup F} \right) = 3 \\\ \end{gathered}
Hence,$P\left( {E \cup F} \right) = \dfrac{{n\left( {E \cup F} \right)}}{{n\left( S \right)}} = \dfrac{3}{4}$

Therefore, the probability of $E \cup F$ is $\dfrac{3}{4}$

Note: There are some main rules associated with basic probability:
A) P(not A)=1-P(A)
B) P(A or B)=P(event A occurs or event B occurs or both occur)
C) P(A and B)=P(both event A and event B occurs)
D) The general Addition Rule: P(A or B) = P(A) + P(B) – P(A and B)