Probability

In a class of 30 students, 17 study Mathematics, 19 study Biology and 11 study both subjects. What is the probability that a student selected at random from the class studies neither subject?

The probability of a student studying neither subject is the complement of the probability of a student studying either or both.

The probability of a student studying either or both is \[ \begin{align} P(\text{Maths or Biology}) &= P(\text{Maths) + P(Biology)} \\ &\quad - P(\text{Maths and Biology}) \\ &= \frac{17}{30} + \frac{19}{30} - \frac{11}{30} \\ &= \frac{25}{30} \end{align} \ \]

Calculate the probability of the complement.

\[ \begin{align} P(\text{Neither}) &= 1 - P(\text{Maths or Biology}) \\ &= 1 - \frac{25}{30} \\ &= \frac{1}{6} \end{align} \ \]

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Now You Try:

In a class of 41 students, 29 study Mathematics, 31 study Biology and 17 study both subjects. What is the probability that a student selected at random from the class studies neither subject?