Mathematics, 11.03.2020 22:22 youcandoit13

Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If X is the number of defective items in a randomly drawn sample of 10 items from the batch, find (a) P{X=0} and (b) P(X>2}.

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Answer from: Joejar63

a) P(X = 0) = 0.5223

b) P(X > 2) = 0.0125

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

The order that the items are selected is not important, so the combinations formula is used to solve this problem.

Combinations formula:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

In this problem, we have that

100 items

6 defective

94 not defective.

a) P{X=0}

None defective.

Desired outcomes

Combinations of 10 from a set of 94. So

$D = C_{94,10} = \frac{94!}{10!84!} = 9041256800000$

Total outcomes:

Combinations of 10 from a set of 100. So

$T = C_{100,10} = \frac{100!}{10!90!} = 17310309000000$

P(X = 0)

$p = \frac{D}{T} = \frac{9041256800000}{17310309000000} = 0.5223$

(b) P(X>2}.

Either two or less are defective, or more than two are defective. The sum of the probabilities of these events is decimal 1. So

$P(X \leq 2) + P(X 2) = 1$

$P(X 2) = 1 - P(X \leq 2)$

In which

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$

P(X = 0)

$P(X = 0) = \frac{D}{T} = \frac{9041256800000}{17310309000000} = 0.5223$

P(X = 1)

Desired outcomes

Combinations of 9 from a set of 94(non defecive) and 1 from a set of 6(defective). So

$D = C_{84,9}*C_{6,1} = \frac{94!}{9!85!}*\frac{6!}{1!5!} = 6382063700000$

Total outcomes:

Combinations of 10 from a set of 100. So

$T = C_{100,10} = \frac{100!}{10!90!} = 17310309000000$

P(X = 1)

$P(X = 1) = \frac{D}{T} = \frac{6382063700000}{17310309000000} = 0.3687$

P(X = 2)

Desired outcomes

Combinations of 8 from a set of 94(non defecive) and 2 from a set of 6(defective). So

$D = C_{94,8}*C_{6,2} = \frac{94!}{8!86!}*\frac{6!}{2!4!} = 1669726000000$

Total outcomes:

Combinations of 10 from a set of 100. So

$T = C_{100,10} = \frac{100!}{10!90!} = 17310309000000$

P(X = 2)

$P(X = 2) = \frac{D}{T} = \frac{1669726000000}{17310309000000} = 0.0965$

Finally

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.5223 + 0.3687 + 0.0965 = 0.9875$

$P(X 2) = 1 - P(X \leq 2) = 1 - 0.9875 = 0.0125$

Answer from: Quest

"it's not the heat, it's the humidity". that's a partly valid phrase you may have heard in the summer, but it's actually both. the heat index, also known as the apparent temperature, is what the temperature feels like to the human body when relative humidity is combined with the air temperature. this has important considerations for the human body's comfort. when the body gets too hot, it begins to perspire or sweat to cool itself off. if the perspiration is not able to evaporate, the body cannot regulate its temperature. evaporation is a cooling process. when perspiration is evaporated off the body, it effectively reduces the body's temperature. when the atmospheric moisture content (i.e. relative humidity) is high, the rate of perspiration from the body decreases. in other words, the human body feels warmer in humid conditions. the opposite is true when the relative humidity decreases because the rate of perspiration increases. the body actually feels cooler in arid conditions. there is direct relationship between the air temperature and relative humidity and the heat index, meaning as the air temperature and relative humidity increase (decrease), the heat index increases (decreases).

step-by-step explanation:

75% of the temperatures are below what value? how do you know? 75% of the temperatures are above w

Answer from: Quest

step-by-step explanation:

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Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If X...