An organisation selected $2400$ families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The information gathered is listed in the table below:
Monthly income (in Rs.) / Vehicles per family $0$ $1$ $2$ Above $2$ Less than $7000$ $10$ $160$ $25$ $0$ $7000 - 10000$ $0$ $305$ $27$ $2$ $10000 - 13000$ $1$ $535$ $29$ $1$ $13000 - 16000$ $2$ $469$ $59$ $25$ $16000$ or more $1$ $579$ $82$ $88$
Suppose a family is chosen. Find the probability that the family chosen is:
$(i)$ Earning Rs. $10000 - 13000$ per month and owning exactly $2$ vehicles.
$(ii)$ Earning Rs. $16000$ or more per month and owning exactly $1$ vehicle.
$(iii)$ Earning less than Rs. $7000$ per month and does not own any vehicle.
$(iv)$ Earning Rs. $13000 - 16000$ per month and owning more than $2$ vehicles.
$(v)$ Owning not more than $1$ vehicle.
| Monthly income (in Rs.) / Vehicles per family | $0$ | $1$ | $2$ | Above $2$ |
| Less than $7000$ | $10$ | $160$ | $25$ | $0$ |
| $7000 - 10000$ | $0$ | $305$ | $27$ | $2$ |
| $10000 - 13000$ | $1$ | $535$ | $29$ | $1$ |
| $13000 - 16000$ | $2$ | $469$ | $59$ | $25$ |
| $16000$ or more | $1$ | $579$ | $82$ | $88$ |
Suppose a family is chosen. Find the probability that the family chosen is:
$(i)$ Earning Rs. $10000 - 13000$ per month and owning exactly $2$ vehicles.
$(ii)$ Earning Rs. $16000$ or more per month and owning exactly $1$ vehicle.
$(iii)$ Earning less than Rs. $7000$ per month and does not own any vehicle.
$(iv)$ Earning Rs. $13000 - 16000$ per month and owning more than $2$ vehicles.
$(v)$ Owning not more than $1$ vehicle.
(N/A) Total number of families surveyed $= 2400$.
$(i)$ Number of families earning Rs. $10000 - 13000$ per month and owning exactly $2$ vehicles $= 29$.
Probability $= \frac{29}{2400}$.
$(ii)$ Number of families earning Rs. $16000$ or more per month and owning exactly $1$ vehicle $= 579$.
Probability $= \frac{579}{2400}$.
$(iii)$ Number of families earning less than Rs. $7000$ per month and owning no vehicle $= 10$.
Probability $= \frac{10}{2400} = \frac{1}{240}$.
$(iv)$ Number of families earning Rs. $13000 - 16000$ per month and owning more than $2$ vehicles $= 25$.
Probability $= \frac{25}{2400} = \frac{1}{96}$.
$(v)$ Number of families owning not more than $1$ vehicle (i.e.,$0$ or $1$ vehicle) $= (10 + 160) + (0 + 305) + (1 + 535) + (2 + 469) + (1 + 579) = 170 + 305 + 536 + 471 + 580 = 2062$.
Probability $= \frac{2062}{2400} = \frac{1031}{1200}$.
$(i)$ Number of families earning Rs. $10000 - 13000$ per month and owning exactly $2$ vehicles $= 29$.
Probability $= \frac{29}{2400}$.
$(ii)$ Number of families earning Rs. $16000$ or more per month and owning exactly $1$ vehicle $= 579$.
Probability $= \frac{579}{2400}$.
$(iii)$ Number of families earning less than Rs. $7000$ per month and owning no vehicle $= 10$.
Probability $= \frac{10}{2400} = \frac{1}{240}$.
$(iv)$ Number of families earning Rs. $13000 - 16000$ per month and owning more than $2$ vehicles $= 25$.
Probability $= \frac{25}{2400} = \frac{1}{96}$.
$(v)$ Number of families owning not more than $1$ vehicle (i.e.,$0$ or $1$ vehicle) $= (10 + 160) + (0 + 305) + (1 + 535) + (2 + 469) + (1 + 579) = 170 + 305 + 536 + 471 + 580 = 2062$.
Probability $= \frac{2062}{2400} = \frac{1031}{1200}$.
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Similar Questions
The following frequency distribution table shows the concentration of sulphur dioxide $(SO_2)$ in the air in parts per million $(ppm)$ of a certain city for $30$ days. Using this table,find the probability of the concentration of sulphur dioxide being in the interval $0.12 - 0.16$ on any of these days.
Concentration of $SO_2$ (in $ppm$) Number of days (frequency) $0.00-0.04$ $4$ $0.04-0.08$ $9$ $0.08-0.12$ $9$ $0.12-0.16$ $2$ $0.16-0.20$ $4$ $0.20-0.24$ $2$ Total $30$
| Concentration of $SO_2$ (in $ppm$) | Number of days (frequency) |
| $0.00-0.04$ | $4$ |
| $0.04-0.08$ | $9$ |
| $0.08-0.12$ | $9$ |
| $0.12-0.16$ | $2$ |
| $0.16-0.20$ | $4$ |
| $0.20-0.24$ | $2$ |
| Total | $30$ |
Medium
View Solution$A$ 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.
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
View Solution$A$ 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.
| Outcome | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
| Frequency | $179$ | $150$ | $157$ | $149$ | $175$ | $190$ |
Find the probability of getting each outcome.
Medium
View SolutionFifty seeds were selected at random from each of $5$ bags of seeds,and were kept under standardised conditions favourable to germination. After $20$ days,the number of seeds which had germinated in each collection were counted and recorded as follows:
Bag $1$ $2$ $3$ $4$ $5$ Number of seeds germinated $40$ $48$ $42$ $39$ $41$
What is the probability of germination of
$(i)$ more than $40$ seeds in a bag?
$(ii)$ $49$ seeds in a bag?
$(iii)$ more than $35$ seeds in a bag?
| Bag | $1$ | $2$ | $3$ | $4$ | $5$ |
| Number of seeds germinated | $40$ | $48$ | $42$ | $39$ | $41$ |
What is the probability of germination of
$(i)$ more than $40$ seeds in a bag?
$(ii)$ $49$ seeds in a bag?
$(iii)$ more than $35$ seeds in a bag?
Medium
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