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
- If one event is unaffected by the outcome of another event, the two events are said to be:
1. Dependent
2. Independent
3. Mutually exclusive
4. Mutually inclusive - Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \dots, 2026\}$.
Which of the following is the probability that the product of all the elements of $M$ is even? - A fair coin is tossed 6 times. The probability of getting at least 5 heads is _________________. (rounded off to two decimal places)
- Consider a random experiment of rolling a fair die. The probability of getting an odd number or a number less than 4 is ______ (Rounded off to two decimal places).
- In a null hypothesis significance test, the significance level $\alpha$ was set to 0.01. For which one or more of the following $p$ values would the null hypothesis be rejected?
