It's been a while since my statistics course and I was wondering if someone could help me with the following question:

All trainers work in the accounting department. 60% of the trainers are bilingual. Joe is a trainer.

What can be validly concluded from the information provided above?

1. Joe is 60% likely to be bilingual
2. Joe is 40% likely to be not bilingual
3. We can't determine whether Joe is bilingual or not
4. Joe is bilingual

I believe that option 3 is the best answer because Joe either is or is not bilingual. I think that the answer would be different if the question asked about a "random employee".

I would write down what you know and what you're looking for.

$\begin{eqnarray*} P(\text{accounting}|\text{trainer}) &=& 1 \\ P(\text{bilingual}|\text{trainer}) &=& 0.6 \\ \end{eqnarray*}$

We want to know $P(\text{Joe is bilingual})$.

I would make the step that, since Joe is a trainer, $P(\text{Joe is bilingual})=P(\text{bilingual}|\text{Joe})=P(\text{bilingual}|\text{trainer})$.

Based on this, I think that 1, 2, and 3 are all valid conclusions.

• Is it really true that $P(\textrm{bilingual | Joe}) = P(\textrm{bilingual | trainer})$? I think what the OP was getting at is this: does it make a difference that a specific employee, Joe, is chosen, rather than a random employee? Joe is either bilingual or not, so can we say that $P(\textrm{bilingual | Joe}) \in \{0,1\}$, whereas $P(\textrm{bilingual | trainer}) = 0.6$? Nov 18, 2015 at 18:44
• Ah, that's a really good point. It would follow the same logic as a confidence interval's probability of containing the parameter must be 0 or 1. Damn these subtle differences! :) Nov 18, 2015 at 19:00
• @Théophile I think you just highlighted the difference between a frequentist and a Bayesian interpretation. Personally, I think the Bayesian interpretation is more sensible here. I.e. what odds are you willing to take to bet money on Joe being bilingual? With which odds are you going to break even in the long run if you keep taking these bets? Nov 8, 2017 at 14:55
• @rinspy That makes sense, but I still have a problem with the wording of the question; I would want to know how Joe was chosen. We don't know what it means to look at the "long run" when we don't know what happened: was a random trainer chosen, and it happened to be Joe, or is Joe always chosen as the representative because he's so friendly? Nov 10, 2017 at 15:30
• @rinspy To put it another way, I see the question like this: "40% of the numbers from 1 to 10 are prime numbers. 8 is a number from 1 to 10. What is the probability that 8 is prime?" Nov 10, 2017 at 15:36