Mutually exclusive events questions with answers

Questions

1. Question 1Add the probabilities

   The probability that a spinner lands on red is $0.2$ and the probability that it lands on blue is $0.1$. Find the probability that it lands on red or blue.

   Show the worked solution

   1. Step 1: Can both happen at once? Choose how to combine the probabilitiesThe two can't both happen: they are mutually exclusive, so add.$P(A \text{ or } B) = P(A) + P(B)$$\mathstrut P = 0.2 + 0.1$$P = 0.3$
2. Question 2Add the probabilities

   The probability that a randomly chosen car is white is $0.2$ and the probability that it is black is $0.45$. Find the probability that it is white or black.

   Show the worked solution

   1. Step 1: Can both happen at once? Choose how to combine the probabilitiesThe two can't both happen: they are mutually exclusive, so add.$P(A \text{ or } B) = P(A) + P(B)$$\mathstrut P = 0.2 + 0.45$$P = 0.65$
3. Question 3Mutually exclusive or not?

   A card is drawn at random from cards numbered $1$ to $20$. State whether the events "an even number" and "less than 4" are mutually exclusive.

   Show the worked solution

   1. Step 1: Mutually exclusive means they can't happen together$2$ is both an even number and less than 4.
   2. Step 2: So the answer isNo.
4. Question 4Mutually exclusive or not?

   A card is drawn at random from cards numbered $1$ to $12$. State whether the events "an even number" and "less than 4" are mutually exclusive.

   Show the worked solution

   1. Step 1: Mutually exclusive means they can't happen together$2$ is both an even number and less than 4.
   2. Step 2: So the answer isNo.
5. Question 5Neither event

   $A$ and $B$ are mutually exclusive events with $P(A) = \dfrac{1}{6}$ and $P(B) = \dfrac{1}{5}$. Find the probability that neither $A$ nor $B$ occurs.

   Show the worked solution

   1. Step 1: P(A or B) for mutually exclusive events$\mathstrut P(A \text{ or } B) = \dfrac{1}{6} + \dfrac{1}{5}$$P(A \text{ or } B) = \dfrac{11}{30}$
   2. Step 2: Neither: what is left of the total probability, 1?$\mathstrut P(\text{neither}) = 1 - \dfrac{11}{30}$$P(\text{neither}) = \dfrac{19}{30}$
6. Question 6Neither event

   $A$ and $B$ are mutually exclusive events with $P(A) = \dfrac{2}{7}$ and $P(B) = \dfrac{1}{4}$. Find the probability that neither $A$ nor $B$ occurs.

   Show the worked solution

   1. Step 1: P(A or B) for mutually exclusive events$\mathstrut P(A \text{ or } B) = \dfrac{2}{7} + \dfrac{1}{4}$$P(A \text{ or } B) = \dfrac{15}{28}$
   2. Step 2: Neither: what is left of the total probability, 1?$\mathstrut P(\text{neither}) = 1 - \dfrac{15}{28}$$P(\text{neither}) = \dfrac{13}{28}$