Probability Calculation for Non-Defective Items
This problem involves calculating the probability of drawing two non-defective thermometers consecutively from a box without replacement.
Identifying Key Values
- Total number of thermometers in the box: 20
- Number of defective thermometers: 3
- Number of non-defective thermometers: $20 - 3 = 17$
Calculating Sequential Probabilities
We need to find the probability that the first thermometer drawn is non-defective AND the second thermometer drawn is also non-defective.
The probability of the first thermometer being non-defective is the ratio of non-defective thermometers to the total number of thermometers:
$P(\text{1st is non-defective}) = \frac{\text{Number of non-defective thermometers}}{\text{Total number of thermometers}} = \frac{17}{20}$
Since the drawing is done *without replacement*, after drawing one non-defective thermometer, there are fewer thermometers left in the box.
- Remaining thermometers: $20 - 1 = 19$
- Remaining non-defective thermometers: $17 - 1 = 16$
The probability of the second thermometer being non-defective, given the first was non-defective, is:
$P(\text{2nd is non-defective} | \text{1st is non-defective}) = \frac{\text{Remaining non-defective thermometers}}{\text{Remaining thermometers}} = \frac{16}{19}$
Overall Probability Calculation
To find the probability that both thermometers drawn are non-defective, we multiply the probabilities of each step:
$P(\text{Both are non-defective}) = P(\text{1st is non-defective}) \times P(\text{2nd is non-defective} | \text{1st is non-defective})$
$P(\text{Both are non-defective}) = \frac{17}{20} \times \frac{16}{19}$
$P(\text{Both are non-defective}) = \frac{17 \times 16}{20 \times 19} = \frac{272}{380}$
Simplifying the fraction gives:
$P(\text{Both are non-defective}) = \frac{68}{95}$
To express this probability as a percentage, we convert the fraction to a decimal and multiply by 100:
$ \text{Probability (decimal)} = \frac{68}{95} \approx 0.715789... $
$ \text{Probability (percent)} = 0.715789... \times 100\% \approx 71.5789...\% $
Rounding to two decimal places, the probability is approximately 71.58%.
Important Questions from Probability
Three dice are thrown. What is the probability of getting a sum which is a perfect square?
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
What is the probability that all three boys sit together?
