Math / Data & Statistics

Question $v \sqrt{44(0.2)(0.8)}$
10. Suppose the probability of a major earthquake on a given day is 1 out of 15,000 . Use the Poisson distribution to approximate the probability that there will be at least one major earthquake in the next 2000 days. poissun 1

Studdy Solution

STEP 1

What is this asking? What's the chance of at least one big earthquake happening in the next 2000 days, if the daily chance is super small? Watch out! We're approximating with the Poisson distribution, not calculating the exact probability!
Also, "at least one" is a sneaky way of asking about something else!

STEP 2

1. Calculate the average number of earthquakes.
2. Find the probability of zero earthquakes.
3. Calculate the probability of at least one earthquake.

STEP 3

We're given that the probability of an earthquake on any given day is $1/15000$ .
We want to find the average number of earthquakes over 2000 days.
This is like saying, "If something happens $1/15000$ of the time, how many times do we expect it to happen in 2000 tries?"

STEP 4

To find the average number of earthquakes ( $\lambda$ ), we multiply the probability of an earthquake on a single day by the number of days: $\lambda = \frac{1}{15000} \cdot 2000 = \frac{2000}{15000} = \frac{2}{15}$ So, we expect an average of $\frac{2}{15}$ earthquakes over 2000 days.
This makes sense, it's a small number because earthquakes are rare!

STEP 5

The Poisson distribution gives us the probability of exactly $k$ events happening, given an average rate of $\lambda$ .
The formula is: $P(k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}$ We want to find the probability of zero earthquakes, so we'll plug in $k = 0$ .

STEP 6

Remember, $\lambda$ is the average number of earthquakes we calculated earlier, which is $\frac{2}{15}$ .
Also, $0! = 1$ and anything raised to the power of 0 is 1.
So, we have: $P(0) = \frac{e^{-\frac{2}{15}} \cdot (\frac{2}{15})^0}{0!} = \frac{e^{-\frac{2}{15}} \cdot 1}{1} = e^{-\frac{2}{15}}$

STEP 7

The probability of at least one earthquake is the same as $1$ minus the probability of no earthquakes.
Think of it like this: all the possible outcomes add up to 1 (or 100%).
If we subtract the chance of no earthquakes, we're left with the chance of one or more earthquakes.

STEP 8

We just calculated the probability of zero earthquakes as $e^{-\frac{2}{15}}$ .
Therefore, the probability of at least one earthquake is: $P(\text{at least one}) = 1 - P(0) = 1 - e^{-\frac{2}{15}}$

STEP 9

The approximate probability of at least one major earthquake in the next 2000 days is $1 - e^{-\frac{2}{15}}$ .

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Question $v \sqrt{44(0.2)(0.8)}$
10. Suppose the probability of a major earthquake on a given day is 1 out of 15,000 . Use the Poisson distribution to approximate the probability that there will be at least one major earthquake in the next 2000 days. poissun 1

Studdy Solution

STEP 1

STEP 2

1. Calculate the average number of earthquakes.
2. Find the probability of zero earthquakes.
3. Calculate the probability of at least one earthquake.

STEP 3

We're given that the probability of an earthquake on any given day is $1/15000$ .
We want to find the average number of earthquakes over 2000 days.
This is like saying, "If something happens $1/15000$ of the time, how many times do we expect it to happen in 2000 tries?"

STEP 4

STEP 5

The Poisson distribution gives us the probability of exactly $k$ events happening, given an average rate of $\lambda$ .
The formula is: $P(k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}$ We want to find the probability of zero earthquakes, so we'll plug in $k = 0$ .

STEP 6

Remember, $\lambda$ is the average number of earthquakes we calculated earlier, which is $\frac{2}{15}$ .
Also, $0! = 1$ and anything raised to the power of 0 is 1.
So, we have: $P(0) = \frac{e^{-\frac{2}{15}} \cdot (\frac{2}{15})^0}{0!} = \frac{e^{-\frac{2}{15}} \cdot 1}{1} = e^{-\frac{2}{15}}$

STEP 7

The probability of at least one earthquake is the same as $1$ minus the probability of no earthquakes.
Think of it like this: all the possible outcomes add up to 1 (or 100%).
If we subtract the chance of no earthquakes, we're left with the chance of one or more earthquakes.

STEP 8

We just calculated the probability of zero earthquakes as $e^{-\frac{2}{15}}$ .
Therefore, the probability of at least one earthquake is: $P(\text{at least one}) = 1 - P(0) = 1 - e^{-\frac{2}{15}}$

STEP 9

The approximate probability of at least one major earthquake in the next 2000 days is $1 - e^{-\frac{2}{15}}$ .

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Questionv44(0.2)(0.8)v \sqrt{44(0.2)(0.8)}v44(0.2)(0.8)​10. Suppose the probability of a major earthquake on a given day is 1 out of 15,000 . Use the Poisson distribution to approximate the probability that there will be at least one major earthquake in the next 2000 days. poissun 1

Studdy Solution

STEP 1

STEP 2

1. Calculate the average number of earthquakes.2. Find the probability of *zero* earthquakes.3. Calculate the probability of at least one earthquake.

STEP 3

We're given that the probability of an earthquake on any given day is 1/150001/150001/15000.We want to find the average number of earthquakes over 2000 days.This is like saying, "If something happens 1/150001/150001/15000 of the time, how many times do we expect it to happen in 2000 tries?"

STEP 4

STEP 5

STEP 6

STEP 7

The probability of *at least one* earthquake is the same as 111 minus the probability of *no* earthquakes.Think of it like this: all the possible outcomes add up to 1 (or 100%).If we subtract the chance of *no* earthquakes, we're left with the chance of *one or more* earthquakes.

STEP 8

We just calculated the probability of zero earthquakes as e−215e^{-\frac{2}{15}}e−152​.Therefore, the probability of at least one earthquake is: P(at least one)=1−P(0)=1−e−215 P(\text{at least one}) = 1 - P(0) = 1 - e^{-\frac{2}{15}} P(at least one)=1−P(0)=1−e−152​

STEP 9

The approximate probability of at least one major earthquake in the next 2000 days is 1−e−2151 - e^{-\frac{2}{15}}1−e−152​.

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Question $v \sqrt{44(0.2)(0.8)}$
10. Suppose the probability of a major earthquake on a given day is 1 out of 15,000 . Use the Poisson distribution to approximate the probability that there will be at least one major earthquake in the next 2000 days. poissun 1

1. Calculate the average number of earthquakes.
2. Find the probability of zero earthquakes.
3. Calculate the probability of at least one earthquake.

We're given that the probability of an earthquake on any given day is $1/15000$ .
We want to find the average number of earthquakes over 2000 days.
This is like saying, "If something happens $1/15000$ of the time, how many times do we expect it to happen in 2000 tries?"

The probability of at least one earthquake is the same as $1$ minus the probability of no earthquakes.
Think of it like this: all the possible outcomes add up to 1 (or 100%).
If we subtract the chance of no earthquakes, we're left with the chance of one or more earthquakes.

We just calculated the probability of zero earthquakes as $e^{-\frac{2}{15}}$ .
Therefore, the probability of at least one earthquake is: $P(\text{at least one}) = 1 - P(0) = 1 - e^{-\frac{2}{15}}$

The approximate probability of at least one major earthquake in the next 2000 days is $1 - e^{-\frac{2}{15}}$ .