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In a school, 55% of the students learn French and 40% learn Spanish…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
In a school, of the students learn French and learn Spanish. Some learn both. The head of languages says that exactly half of the Spanish learners also learn French. A student is chosen at random, and the school wants to plan timetables from the following four numbers. Let be the event that the student learns French and the event that the student learns Spanish. Find the probability that the student learns both languages. Find the probability that the student learns at least one of the two. Find the probability that the student learns Spanish, given that the student learns French. Then find the probability that the student does not learn French, given that the student does not learn Spanish. Give decimals to four places where needed.
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The worked solution
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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. Both languages. Half of the Spanish learners also learn French, so .
Step 2. At least one. .
Step 3. Spanish given French. .
Step 4. Not French given not Spanish. The students who learn neither language are of the school. Those who do not learn Spanish are . So .
Answer: , , about and about .
Check. The events are not independent. Given Spanish, the chance of French is , which is below . In the other direction, is below . The two comparisons point the same way, as they should.
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