Q.4.64 - A First Course in Probability

Question

The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month. 

(a) Find the probability that in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in a given month.

(b) What is the probability that there will be at least 2 months during the year that will have 8 or more suicides? 

(c) Counting the present month as month number 1, what is the probability that the first month to have 8 or more suicides will be month number $i,i\ge 1$? What assumptions are you making? 

Step-by-Step Solution

Answer

$\left(a\right)$The probability that in a city of $400,000$ inhabitants within this state, there will be $8$ or more suicides in a given month is $0.05113$.

$\left(b\right)$The probability that there will be at least $2$ months during the year that will have $8$ or more suicides is$0.125$.

$\left(c\right)$The probability that the first month to have 8 or more suicides will be month number $i,i\ge 1$is $P(Z=i)=(1-p{)}^{i-1}·p$

Step 1: Given Information (Part a)

The suicide rate in a certain state is $1$ suicide per $100,000$ inhabitants per month. 

Step 2: Calculation (Part a)

Since we are given that there is a rate of $1$ suicide per $100,000$ inhabitants, we hold that there is the rate of $4$ suicides per $400,000$ inhabitants. So, the number of suicides is can be expressed as a random variable $X$ with allocation $X~Pois\left(4\right)$.

So,

$P(X\ge 8)=1-\sum _{j=0}^{7}P(X=j)=1-\sum _{j=0}^7\frac{4^j}{j!}{e}^{-4}\approx 0.05113$

Step 3: Final answer (Part a)

The probability of  given city found to be $0.05113$.

Step 4: Given information (Part b)

The suicide rate in a certain state is $1$ suicide per $100,000$ inhabitants per month.

Step 5: Calculation (Part b)

Every month has the probability of $p=0.05113$ that there will be more or similar to eight suicides that month. If we mark $Y$ as the random variable that marks the number of months that have that property, it has approximately Poisson distribution with parameter $12·0.05113=0.61$.

Hence

$P(Y\ge 2)=1-P(Y=0)-P(Y=1)=1-e^{-0.61}-0.61e^{-0.61}\approx 0.125$

Step 6: Final answer(Part b)

The probability of suicides found to be $0.125$.

Step 7: Given information (Part c)

The suicide rate in a certain state is $1$ suicide per $100,000$ inhabitants per month. 

Step 8: Calculation (Part c)

Define random variable $Z$ that marks the first month in which there were $8$ or more suicides. Observe that because of the nature of the problem, We have that $Z~Geom\left(p\right)$, where $p$ is the probability calculated in $\left(a\right)$.

So,

$P(Z=i)=(1-p{)}^{i-1}·p$

Step 9: Final answer (Part c)

The assumption of probability found to be $P(Z=i)=(1-p{)}^{i-1}·p$