A box holds 240 eggs. The probability that an egg is brown is 0.05. How do you find the probability that there are at least 10 brown eggs in the box?
Let
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To find the probability that there are at least 10 brown eggs in the box, we can use the binomial probability formula. The probability of getting exactly (k) successes (brown eggs) in (n) trials (total number of eggs) is given by the formula:
[ P(X = k) = \binom{n}{k} \times p^k \times (1 - p)^{n - k} ]
Where:
- ( n ) is the total number of trials (eggs in the box), which is 240 in this case.
- ( k ) is the number of successes (brown eggs), which we want to be at least 10.
- ( p ) is the probability of success (probability that an egg is brown), which is 0.05.
- ( \binom{n}{k} ) represents the number of ways to choose ( k ) successes out of ( n ) trials, also known as the binomial coefficient.
To find the probability that there are at least 10 brown eggs, we need to sum the probabilities of getting 10, 11, 12, ..., up to 240 brown eggs. This can be computationally intensive. However, we can use a shortcut by finding the probability of getting fewer than 10 brown eggs and subtracting it from 1 to find the probability of getting at least 10 brown eggs:
[ P(X \geq 10) = 1 - P(X < 10) ]
Now we can calculate ( P(X < 10) ) using the binomial probability formula:
[ P(X < 10) = P(X = 0) + P(X = 1) + \ldots + P(X = 9) ]
After finding ( P(X < 10) ), we subtract it from 1 to get ( P(X \geq 10) ), the probability of having at least 10 brown eggs in the box.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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