Uniform Distribution - Is my solution correct?

Question

I'm preparing for an exam in Probability and Statistics and since I am self-studying, I don't have answers to refer to. So would be really great if someone would please take the time and check if my solution for the following problem is correct. Thank you in advance!

X corresponds to the duration of Mary's commute to work, and Y to the duration of John's. X is uniformly distributed in [15,25] and Y in [15,30]. X and Y are assumed to be independent. What is the probability that Mary and John both take more than 20 minutes to get to work?

So we are looking for: P(X>20,Y>20)=P(X>20)xP(Y>20). In other words we are looking for: P[1-(P(X≦20),1-(P(Y≦20).

As per the rule P with Uniform Distribution: P([c,d])= $\int_c^d\frac{1}{b-a}dx=\frac{d-c}{b-a}$

So in our cases:

1. $P(X≦20)= P([15,20])= \int_{15}^{20}\frac{1}{25-15}dx=\frac{20-15}{25-15}=0.5$ from here follows that $1-P(X≦20)= 1-0.5=0.5$

So $P(X>20)=0.5$

2. $P(Y≦20)= P([15,20])= \int_{15}^{20}\frac{1}{30-15}dx=\frac{20-15}{30-15}=0.33$ from here follows that $1-P(Y≦20)= 1-0.33=0.67$

So $P(Y>20)=0.57$

Going back to

P(X>20,Y>20)=P(X>20)xP(Y>20)=0.5x0.67=0.335

In conclusion, that probability that both Mary and John take more than 20 minutes to get to work is 0.335.

Answer 1

Generally looks good. I wouldn’t introduce decimals and start rounding—the exact answer is $1/3$, which is a bit different from $0.335$.

Also I’m not sure whether the notation $P([a,b])$ is standard. I typically write $P(a\leq{X}\leq{b})$, but that might be my background.

Also I’m not sure what you meant here:

P[1-(P(X≦20),1-(P(Y≦20)

I think you meant $$[1-P(X\leq20)]\cdot[1-P(Y\leq20)],$$ so be careful with notation.

Answer 2

To check this intuitively, You could as well calculated it easily $P(X>20)$ and $P(Y>20)$. Uniform distribution means every event is equally likely So for X commute time is uniformly distributed between 15 and 25. So the probability of reaching exactly at a particular minute between 15 and 25 is $1/10$. Similarly for Y it is $1/15$. Now how many minutes above 20 are there for X and Y in their respective ranges? 5 for X and 10 for Y. So, $P(X> 20)=5/10=1/2$ and $P(Y>20)=10/15=2/3$. Since the events are independent, $P(X>20, Y>20)= 1/2 *2/3 =1/3$. If you look at the Probablity graph of the distribution, the integral that you posted is nothing but the rectangular area between $c=21$ and $d=25$ for X and $c=21$ and $d=30$ for Y. The height is the probablity which is $1/10$ and $1/15$ Respectively. Since it’s the same for every event you form a nice $y=f(x) = 1/(b-a)$ line.Now if the distribution is not Uniform, then each such event would have differing probability and so the integral area under the curve will vary.