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%.