Two events are **independent** if the outcome of each event does not affect the outcome of the other event. For example, when we roll a die, there is always a \dfrac{1}{6} chance that it will land on 3. The outcome of the first roll will have no effect on outcome of the second roll.

The probability of a single event is the ratio of the number of favorable outcomes to the total number of outcomes:\text{Probability} = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

The probability of two independent events is given by the formula:P\left(A\text{ and } B\right)=P\left(A\right) \cdot P\left(B\right)This means that the probability of both A and B occurring can be found by multiplying the probabilities of each event occurring separately.

Tree diagrams can be helpful for calculating the probabilities to two events. We have seen that tree diagrams can be used to list the sample space of an experiment. However, it can be difficult to list the outcomes in a simple tree diagram when there are many outcomes.

Consider the tree diagrams shown.

Could these tree diagrams represent the same situation? Explain.

For the tree diagram without probabilities on the branches, how could you find the probability of CC?

Knowing the probability of CC from question 2, how could you find the probability of CC from the other tree diagram?

What are the benefits and drawbacks of each of the diagrams?

In a **probability tree diagram**, the probability of each outcome is written on the branches.

Here is an example of an experiment with 2 events.

When using a probability tree diagram to find the probability of an outcome that includes both events, we must multiply across the branches of the tree diagram. For example, the probability that a child returns home during the day for both days is\begin{aligned}P\left(D\text{ and }D\right)&=P\left(D\right)\cdot P\left(D\right)\\&=0.7 \cdot 0.7 \\&= 0.49\end{aligned}The probability a child returns home at night on the first day and during the day on the second day is \begin{aligned}P\left(N\text{ and }D\right)&=P\left(N\right)\cdot P\left(D\right)\\&=0.3 \cdot 0.7\\&= 0.21\end{aligned}

When we add the probabilities of all the possible outcomes, it should be equal to 1 (or 100\%).

A standard six-sided die is rolled 691 times.

If it lands on a six 108 times, what is the probability that the next roll will land on a six?

Worked Solution

Two events A and B are such that:

- P\left(A\right)=0.5
- P\left(B\right)=0.7
- P\left(A\text{ and } B\right)=0.3

Determine whether events A and B are independent.

Worked Solution

A standard deck of 52 cards has four queens. Ilham shuffles the deck then selects a card. He puts the card back in the deck, then selects a second card.

Let Q represent the event of drawing a queen and N represent the event of not drawing a queen. The tree diagram shows all the possible outcomes and probabilities:

a

Find the probability that both cards are queens.

Worked Solution

b

Find the probability that neither card is a queen.

Worked Solution

c

Find the probability that Ilham selects only one queen.

Worked Solution

Idea summary

Two events are independent if the outcome of the each event does not affect the outcome of the other event.

If two events are independent, the probability of both events occurring is: P\left(A \text{ and } B\right) = P\left(A\right) \cdot P\left(B\right)This formula can also be used to check if two events are independent.