College school graduate job probability question

Question

The probability that Big Co. will hire new graduates from State U. who have interviews in any given year is 10%. Of the graduates this year, 4 were chosen at random for interviews. What is the probability that exactly 3 of them will be hired? 

Answer 1

In order to answer the question, "What is the probability that EXACTLY 3 of [the 4 students] will be hired?" let's see how many ways this EXACT situation could happen.  

I think visualizing this question will help. Below I have arranged all of the possible ways that exactly 3 of the 4 students could get hired. Let's call them students A, B, C, D. The letter "H" below that student will stand for "Hired." An "X" will mean that they weren't hired.

              Students:     A   B   C   D

first scenario           X   H   H  H

second scenario      H   X   H  H

third scenario         H   H   X  H

fourth scenario       H  H   H   X

In each above scenario, 1 student was NOT hired, and 3 WERE hired. We have now exhausted all of the possible scenarios that could result in exactly 3 of the 4 students getting hired.

It is stated in the original question that each of the 4 students chosen individually have a 10% chance of being hired.

Therefore, the chance that a particular individual student will NOT be hired is (100% – 10%), which is 90% or (0.9 in decimal form).

So when you see an "H" in the above table, think of it representing (0.1)

When you see an "X" in the above table, it represents a chance of (0.9)

To find the likelihood of one of the above particular scenarios happening, multiply each of the probabilities in that scenario. For example, to find the probability that the first scenario would happen (XHHH), multiply the following: (0.9)(0.1)(.01)(0.1), which equals (0.0009) or 0.09%

Since each of the following scenarios 2 through 4 also have a (0.0009) or 0.09% chance of happening, we add the likelihoods of each scenario, #1 through #4, to get the overall probability that EXACTLY three students get hired. By adding up each of the four scenarios, we have covered all of our bases on how exactly three of the four students could get hired:  

The sum of the probabilities of scenarios # 1 through #4:

0.09% + 0.09% + 0.09% + 0.09% = 0.36%

Answer: Therefore, there is a 0.36% (or 0.0036 in decimal form) chance that EXACTLY 3 of the 4 students will be hired.