Two coins are tossed simultaneously $500$ times,and we get:
Two heads : $105$ times
One head : $275$ times
No head : $120$ times
Find the probability of occurrence of each of these events.

$A$ teacher wanted to analyze the performance of two sections of students in a mathematics test of $100$ marks. Looking at their performances,she found that a few students got under $20$ marks and a few got $70$ marks or above. So she decided to group them into intervals of varying sizes as follows: $0-20, 20-30, ..., 60-70, 70-100$. Then she formed the following table:

Marks	Number of students
$0-20$	$7$
$20-30$	$10$
$30-40$	$10$
$40-50$	$20$
$50-60$	$20$
$60-70$	$15$
$70$ and above	$8$
Total	$90$

$(i)$ Find the probability that a student obtained less than $20\%$ in the mathematics test.
$(ii)$ Find the probability that a student obtained marks $60$ or above.

The distance (in $km$) of $40$ engineers from their residence to their place of work were found as follows.

$5$	$3$	$10$	$20$	$25$	$11$	$13$	$7$	$12$	$31$
$19$	$10$	$12$	$17$	$18$	$11$	$32$	$17$	$16$	$2$
$7$	$9$	$7$	$8$	$3$	$5$	$12$	$15$	$18$	$3$
$12$	$14$	$2$	$9$	$6$	$15$	$15$	$7$	$6$	$12$

What is the empirical probability that an engineer lives:
$(i)$ less than $7 \, km$ from her place of work?
$(ii)$ more than or equal to $7 \, km$ from her place of work?
$(iii)$ within $\frac{1}{2} \, km$ from her place of work?

$A$ die is thrown $1000$ times with the frequencies for the outcomes $1, 2, 3, 4, 5$ and $6$ as given in the following table:

Outcome	$1$	$2$	$3$	$4$	$5$	$6$
Frequency	$179$	$150$	$157$	$149$	$175$	$190$

Find the probability of getting each outcome.

$A$ coin is tossed $1000$ times with the following frequencies:
Head : $455$ Tail : $545$
Compute the probability for each event.

Note the frequency of two-wheelers, three-wheelers, and four-wheelers passing by your school gate during a specific time interval. Calculate the probability that any one vehicle chosen from the total vehicles observed is a two-wheeler.