# What’s the increased probability for a masters student given that there are 3 lottery attempts

#### Jeny George

##### New member
I am fairly new to statistics and probability, I'm currently studying for a career change. I was asked this question but I don't know how to navigate through and get to any kind of an answer.

Previously, the government conducted a lottery to award visas to 20,000 advanced-degree holders first. Those who weren't chosen then got a second chance with the other H-1B petitions in a larger 65,000-visa lottery. This year, instead of conducting the advanced-degree lottery first, USCIS will run the regular H-1B cap lottery to meet the 65,000-visa quota with all advanced-degree earners included. It will then put the remaining applicants with U.S. master's degrees or higher into the separate lottery for 20,000 visas.

What’s the increased probability for a masters student given that there are 3 lottery attempts?

Assume 190000 total applicants with 90000 being masters students

Is there a possible explanation for this?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
I am fairly new to statistics and probability, I'm currently studying for a career change. I was asked this question but I don't know how to navigate through and get to any kind of an answer.

Previously, the government conducted a lottery to award visas to 20,000 advanced-degree holders first. Those who weren't chosen then got a second chance with the other H-1B petitions in a larger 65,000-visa lottery. This year, instead of conducting the advanced-degree lottery first, USCIS will run the regular H-1B cap lottery to meet the 65,000-visa quota with all advanced-degree earners included. It will then put the remaining applicants with U.S. master's degrees or higher into the separate lottery for 20,000 visas.

What’s the increased probability for a masters student given that there are 3 lottery attempts?

Assume 190000 total applicants with 90000 being masters students

Is there a possible explanation for this?
Hi Jeny George, welcome to MHB!

Let's take a look at the old system.

Old system for holder of master's degree

The probability that a holder of a master's degree gets a visa in the 20000 lottery is:
$$P(\text{visa in advanced degree lottery}) = \frac{\text{Number awarded}}{\text{Number participants}} = \frac{20\,000}{90\,000} = \frac 29$$
The probability they do not get chosen in this first lottery is then $\frac 79$.

The ones that were not chosen get a second chance.
There are $90\,000-20\,000=70\,000$ that were not chosen. So in the general lottery each of them has another chance of:
$$P(\text{visa in general lottery}) = \frac{\text{Number awarded}}{\text{Number participants}} = \frac{65\,000}{190\,000 + 70\,000} = \frac{1}{4}$$

If the master got a visa in the first lottery it stops.
However, there is a chance he did not get it, and if that chance applies he gets a chance in the general lottery.
The total chance for a master to get a visa in the old system is then:
$$\begin{array}{lcl}P(\text{master gets visa}) &=&& P(\text{visa in advanced degree lottery})\\ &&+& P(\text{did not get visa in advanced degree lottery AND gets visa in general lottery})\\ &=&& P(\text{visa in advanced degree lottery})\\ &&+& P(\text{did not get visa in advanced degree lottery})\cdot P(\text{visa in general lottery})\\ &=&& \frac 29 + \frac 79 \cdot \frac 14 \end{array}$$

Next is to do the same for the new system to find the increase in probability.

Btw, it's not clear to me yet what is meant that there are 3 lottery attempts.
Aren't there 2 lottery attempts in both the old and the new system?
Can you clarify?

#### Jeny George

##### New member
I sincerely apologize for the late reply. To clarify your doubt, there is a 3 year OPT period after your done with your masters. So 1 attempt for each year which is what is meant by 3 attempts above.