When two dice are thrown simultaneously,find the probability of the following:
$(1)$ The numbers are the same.
$(2)$ The difference between the numbers is $1$.

$A$ special lottery is to be held to select a student who will live in the only deluxe room available in a hostel. $100$ $III$ year,$150$ $II$ year,and $200$ $I$ year students have applied for the room. Each $III$ year student's name is placed in the lottery $3$ times,each $II$ year student's name $2$ times,and each $I$ year student's name $1$ time. The probability that a $III$ year student gets the room is:

Out of $100$ students,two sections of $40$ and $60$ are formed. If you and your friend are among the $100$ students,what is the probability that you both enter different sections?

Let the sum of two positive integers be $24$. If the probability,that their product is not less than $\frac{3}{4}$ times their greatest positive product,is $\frac{m}{n}$,where $\operatorname{gcd}(m, n)=1$,then $n-m$ equals :

$A$ game consists of tossing a coin $3$ times and noting its outcome. $A$ boy wins the game if all tosses give the same outcome (i.e.,three heads or three tails) and loses the game otherwise. The probability that the boy loses the game is

If there are $4$ red,$3$ pink,$5$ yellow,and $8$ white roses in a garden,what is the probability that a blind person touches a red or a white rose?