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Delicious77 [7]
2 years ago
11

Consider two people being randomly selected. (For simplicity, ignore leap years.)

Mathematics
1 answer:
Setler79 [48]2 years ago
8 0

Answer:

A)144/133225

B)1/365

Step-by-step explanation:

A) We want to find the probability that two people have a birthday on the 18th of any month.

If we Neglect leap year like the question says, it means that there are 365 days in a year.

Now, we have 12 possible number of 18th day in all months that make a calendar year.

Thus;

If two people have birthday on 18th, the probability that the two of them will have birthday on the 18th of any month is;

P(both to have birthday on the 18th of any month) = (12/365) × (12/365) = 144/133225

B) We want to find the probability that two people have a birthday on the same day of the same month. This can be written as;

P(2 people have a birthday on the same day of the same month) + P(2 people do not having birthday on same day of same month) = 1

Probability of 2 people not having birthday on same day of same month = 365/365 × 364/365 = 364/365

P(2 people have a birthday on the same day of the same month) = 1 - (364/365) = 1/365

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According to the article "Characterizing the Severity and Risk of Drought in the Poudre River, Colorado" (J. of Water Res. Plann
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Answer:

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(b) The probability that the length of a drought exceeds its mean value by at least one standard deviation is 0.2064.

Step-by-step explanation:

The random variable <em>Y</em> is defined as the number of consecutive time intervals in which the water supply remains below a critical value <em>y₀</em>.

The random variable <em>Y</em> follows a Geometric distribution with parameter <em>p</em> = 0.409<em>.</em>

The probability mass function of a Geometric distribution is:

P(Y=y)=(1-p)^{y}p;\ y=0,12...

(a)

Compute the probability that a drought lasts exactly 3 intervals as follows:

P(Y=3)=(1-0.409)^{3}\times 0.409=0.0844279\approx0.0844

Thus, the probability that a drought lasts exactly 3 intervals is 0.0844.

Compute the probability that a drought lasts at most 3 intervals as follows:

P (Y ≤ 3) =  P (Y = 0) + P (Y = 1) + P (Y = 2) + P (Y = 3)

              =(1-0.409)^{0}\times 0.409+(1-0.409)^{1}\times 0.409+(1-0.409)^{2}\times 0.409\\+(1-0.409)^{3}\times 0.409\\=0.409+0.2417+0.1429+0.0844\\=0.8780

Thus, the probability that a drought lasts at most 3 intervals is 0.8780.

(b)

Compute the mean of the random variable <em>Y</em> as follows:

\mu=\frac{1-p}{p}=\frac{1-0.409}{0.409}=1.445

Compute the standard deviation of the random variable <em>Y</em> as follows:

\sigma=\sqrt{\frac{1-p}{p^{2}}}=\sqrt{\frac{1-0.409}{(0.409)^{2}}}=1.88

The probability that the length of a drought exceeds its mean value by at least one standard deviation is:

P (Y ≥ μ + σ) = P (Y ≥ 1.445 + 1.88)

                    = P (Y ≥ 3.325)

                    = P (Y ≥ 3)

                    = 1 - P (Y < 3)

                    = 1 - P (X = 0) - P (X = 1) - P (X = 2)

                    =1-[(1-0.409)^{0}\times 0.409+(1-0.409)^{1}\times 0.409\\+(1-0.409)^{2}\times 0.409]\\=1-[0.409+0.2417+0.1429]\\=0.2064

Thus, the probability that the length of a drought exceeds its mean value by at least one standard deviation is 0.2064.

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