Two probability questions

MacDanger

Hi

Doing some stats homework and I have a couple of questions which hopefully someone can help me with.

First question:
Tom sells papers each morning at a stall. The rates of sales corresponds to a Poisson distribution with rate 3/minute.

(a) What is probability he will sell no newspapers in a 90 second period?
(b) Suppose on a given morning he has just sold a paper. What is the probability he will have to wait at least 90 seconds to sell the next?

I'm calculating a) using:

Pr(X = x) = (e-λt)(λt)x / x!

And subbing for λ = 3, t =1.5 (i.e. 90s = 1.5min) and x = 0, which gives a result of 0.011108997.

It’s the second part I’m having the problem with – if he has just sold a paper, isn’t the probability that he has to wait at least 90s to sell the next the same as selling no papers in 90s??

Or am I missing something here??

Second question:

The probability a citizen will die in his twentieth year is 0.00178. A life insurance company sells a $1000 one-year term policy to a 19-year old for $5

What is the expected gain to the insurance company from selling one such policy, ignoring expenses of selling and administration?

I calculated this as:

Expected cost to the company: (1000) (0.00178) = $1.78
Expected benefit: $5
Overall expected gain: $3.22

Is this approach too simplistic??

Thanks in advance

Yakuza

MacDanger wrote: »

Hi
And subbing for λ = 3, t =1.5 (i.e. 90s = 1.5min) and x = 0, which gives a result of 0.011108997.

Looks good

MacDanger wrote: »

It’s the second part I’m having the problem with – if he has just sold a paper, isn’t the probability that he has to wait at least 90s to sell the next the same as selling no papers in 90s??

You're quite correct, the Poisson distribution is memoryless, so resetting the clock after a paper is sold, the probability of zero papers sold in both 90 second time periods is the same.

MacDanger wrote: »

Second question:

The probability a citizen will die in his twentieth year is 0.00178. A life insurance company sells a $1000 one-year term policy to a 19-year old for $5

Expected cost to the company: (1000) (0.00178) = $1.78
Expected benefit: $5
Overall expected gain: $3.22

Again, looks good to me. Insurance companies love insuring young people

MacDanger

Thanks for that