Question

You have decided to sell an item on craigslist. You have given no price in your ad: instead, you ask the interested people reading the ad to submit an offer. Their offers will be uniformly distributed between 0 and 100 dollars (decimals included 1 – hence, an offer of $9.83 is perfectly valid). You would like to make at least $75 from the item. In addition, you receive exactly one offer every day since posting the ad. Answer the following questions. (a) What is the probability a random offer you receive is higher than or equal to $75? (b) Consider you will not accept any offers below $75, what is the probability you sell your item on exactly the third day (so at the third offer received)? (c) You decide not to look at your email for 10 days. When opening your email again to check on the (10) offers you have received, what is the probability that at least one of them offers you $75 or more?

Answer 1

a) 25%

b) 12.5%

c) 99.9%

Explanation:

a)

If we divide the interval [0,100] if four parts of the same length we would obtain the intervals

[0,25], [25,50], [50,75] and [75,100]

Since the offers will be uniformly distributed between 0 and 100 dollars, a given offer has 25% probability of falling in the interval [75,100] (the interval of interest)

b)

Let's call

N= “the offer is less than $75”

Y= “the offer is greater or equal than 75”

In 3 days we have 2*2*2=8 possible outcomes

(N,N,N), (N,N,Y), (N,Y,N), (Y,N,N), (N,Y,Y), (Y,N,Y), (Y,Y,N), (Y,Y,Y).

So the probability of (N,N,Y) is 1/8 = 0.125 = 12.5%

c)

After 10 days we have
`2^(10)=1,024` possible outcomes.

In all this offers, 1,023 are of interest and only one is not (none of the offers is $75 or more).

So, the probability that at least one of them offers you $75 or more is

1,023/1,024 = 0.999 or 99.9%