$A$ 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.
| 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.
- A$\frac{7}{39}, 53/90$
- B$\frac{27}{90}, 69/90$
- C$\frac{7}{19}, 23/90$
- D$\frac{7}{90}, 23/90$
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Similar Questions
An insurance company selected $2000$ drivers at random in a particular city to find a relationship between age and accidents. The data obtained are given in the following table:
Age of drivers (in years) $0$ accidents $1$ accident $2$ accidents $3$ accidents Over $3$ accidents $18-29$ $440$ $160$ $110$ $61$ $35$ $30-50$ $505$ $125$ $60$ $22$ $18$ Above $50$ $360$ $45$ $35$ $15$ $9$
Find the probabilities of the following events for a driver chosen at random from the city:
$(i)$ Being $18-29$ years of age and having exactly $3$ accidents in one year.
$(ii)$ Being $30-50$ years of age and having one or more accidents in a year.
$(iii)$ Having no accidents in one year.
| Age of drivers (in years) | $0$ accidents | $1$ accident | $2$ accidents | $3$ accidents | Over $3$ accidents |
| $18-29$ | $440$ | $160$ | $110$ | $61$ | $35$ |
| $30-50$ | $505$ | $125$ | $60$ | $22$ | $18$ |
| Above $50$ | $360$ | $45$ | $35$ | $15$ | $9$ |
Find the probabilities of the following events for a driver chosen at random from the city:
$(i)$ Being $18-29$ years of age and having exactly $3$ accidents in one year.
$(ii)$ Being $30-50$ years of age and having one or more accidents in a year.
$(iii)$ Having no accidents in one year.
Medium
View SolutionConsider the frequency distribution table which gives the weights of $38$ students of a class.
Weights (in $kg$) Number of students $31-35$ $9$ $36-40$ $5$ $41-45$ $14$ $46-50$ $3$ $51-55$ $1$ $56-60$ $2$ $61-65$ $2$ $66-70$ $1$ $71-75$ $1$ Total $38$
$(i)$ Find the probability that the weight of a student in the class lies in the interval $46-50 \, kg$.
$(ii)$ Give two events in this context,one having probability $0$ and the other having probability $1$.
| Weights (in $kg$) | Number of students |
| $31-35$ | $9$ |
| $36-40$ | $5$ |
| $41-45$ | $14$ |
| $46-50$ | $3$ |
| $51-55$ | $1$ |
| $56-60$ | $2$ |
| $61-65$ | $2$ |
| $66-70$ | $1$ |
| $71-75$ | $1$ |
| Total | $38$ |
$(i)$ Find the probability that the weight of a student in the class lies in the interval $46-50 \, kg$.
$(ii)$ Give two events in this context,one having probability $0$ and the other having probability $1$.
Medium
View Solution$A$ tyre manufacturing company kept a record of the distance covered before a tyre needed to be replaced. The table shows the results of $1000$ cases.
Distance (in $km$) less than $4000$ $4000$ to $9000$ $9001$ to $14000$ more than $14000$ Frequency $20$ $210$ $325$ $445$
If you buy a tyre of this company,what is the probability that:
$(i)$ it will need to be replaced before it has covered $4000 \, km$?
$(ii)$ it will last more than $9000 \, km$?
$(iii)$ it will need to be replaced after it has covered somewhere between $4000 \, km$ and $14000 \, km$?
| Distance (in $km$) | less than $4000$ | $4000$ to $9000$ | $9001$ to $14000$ | more than $14000$ |
| Frequency | $20$ | $210$ | $325$ | $445$ |
If you buy a tyre of this company,what is the probability that:
$(i)$ it will need to be replaced before it has covered $4000 \, km$?
$(ii)$ it will last more than $9000 \, km$?
$(iii)$ it will need to be replaced after it has covered somewhere between $4000 \, km$ and $14000 \, km$?
Medium
View SolutionTwo 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.
Two heads : $105$ times
One head : $275$ times
No head : $120$ times
Find the probability of occurrence of each of these events.
Medium
View SolutionThe 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?
| $5$ | $3$ | $10$ | $20$ | $25$ | $11$ | $13$ | $7$ | $12$ | $31$ |
| $19$ | $10$ | $12$ | $17$ | $18$ | $11$ | $32$ | $17$ | $16$ | $2$ |
| $7$ | $9$ | $7$ | $8$ | $3$ | $5$ | $12$ | $15$ | $18$ | $3$ |
| $12$ | $14$ | $2$ | $9$ | $6$ | $15$ | $15$ | $7$ | $6$ | $12$ |
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?
Medium
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