Probability Calculation: Non-Defective Screws
This problem involves calculating the probability of selecting two non-defective screws sequentially without replacement from a given set.
- Total number of screws = 10
- Number of defective screws = 3
- Number of non-defective screws = 10 - 3 = 7
Step-by-Step Probability Calculation
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First Draw Probability: Calculate the probability that the first screw selected is non-defective.
There are 7 non-defective screws out of a total of 10 screws.
$ P(\text{1st screw is non-defective}) = \frac{\text{Number of non-defective screws}}{\text{Total screws}} = \frac{7}{10} $
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Second Draw Probability: Calculate the probability that the second screw selected is also non-defective, given the first was non-defective. Since the draw is without replacement, there are now 9 screws remaining, and 6 of them are non-defective.
$ P(\text{2nd screw is non-defective} | \text{1st was non-defective}) = \frac{\text{Remaining non-defective screws}}{\text{Remaining total screws}} = \frac{6}{9} $
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Combined Probability: The probability that both screws are non-defective is the product of the probabilities calculated in the previous steps.
$ P(\text{Neither screw is defective}) = P(\text{1st non-defective}) \times P(\text{2nd non-defective} | \text{1st non-defective}) $
$ P(\text{Neither screw is defective}) = \frac{7}{10} \times \frac{6}{9} = \frac{42}{90} $
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Simplification: Simplify the final fraction. The greatest common divisor of 42 and 90 is 6.
$ \frac{42}{90} = \frac{42 \div 6}{90 \div 6} = \frac{7}{15} $
The fraction $\frac{7}{15}$ is equivalent to $\frac{14}{30}$.
The probability that neither of the two screws drawn is defective is $\frac{7}{15}$, which matches the option $\frac{14}{30}$.
Important Questions from Basics of Probability
Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ______ . (rounded off to three decimal places)
- A quadratic polynomial $(x - \alpha)(x - \beta)$ over complex numbers is said to be square invariant if $(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random.
The probability that the roots of the chosen polynomial are equal is ____________ (rounded off to one decimal place) - If, out of 50 fish of a pond, 12 had no ectoparasite and remaining had varying numbers of parasites on them, then how many of fish had at least one parasite ?
For a distribution with mean, median, mode and standard deviation 25, 24, 26 and 5 respectively, Karl Pearson's coefficient of skewness equals to:
Suppose two distributions have the same mean, same standard deviation and are equally skewed, but the first distribution is more peaked than the
other. This feature of the frequency distributions in which they are different are called:
