Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A made-up survey of students counts who learns French, German or Spanish. learn French, German and Spanish. learn French and German, French and Spanish, and German and Spanish. students learn all three. Find how many students learn at least one of the languages. Then find the probability that a student picked at random learns none, and the probability that the student learns exactly one language.
. For any two events . If and are mutually exclusive, the overlap is .
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. At least one. students. The students in all three were added three times and taken away three times, so they are added back once.
Step 2. None. students, so .
Step 3. Exactly one. French only: . German only: . Spanish only: . The total is , so .
Answer: students, and .
Check. Exactly two languages: students. Then , which matches Step 1.
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