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Update Oceanverse.md
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Gitansh-Bansal authored Jun 14, 2024
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Expand Up @@ -1403,24 +1403,24 @@ Thus, the probability of a dart hitting the shaded region is \( \frac{3}{4} \) o

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65.A factory produces items, and each item is independently defective with probability
0.02. If 100 items are produced in a day, what is the expected number of defective items?
0.2. If 100 items are produced in a day, what is the expected number of defective items?
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<summary>Sol.</summary>

To find the expected number of defective items produced in a day by the factory, we use the concept of expectation in probability theory.

Given:
- Each item is defective with probability \( p = 0.02 \).
- Each item is defective with probability \( p = 0.2 \).
- The number of items produced in a day \( n = 100 \).

The expected number of defective items \( E(X) \) can be calculated using the formula for the expectation of a binomial distribution:
\[ E(X) = n \cdot p \]

Substituting the given values:
\[ E(X) = 100 \cdot 0.02 \]
\[ E(X) = 2 \]
\[ E(X) = 100 \cdot 0.2 \]
\[ E(X) = 20 \]

Thus, the expected number of defective items produced in a day is 2.
Thus, the expected number of defective items produced in a day is 20.

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66.A point is chosen at random inside a sphere of radius R. What is the probability that this point is closer to the center of the sphere than to its surface?
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