2.6 Probability Rules

1. What is the probability getting 3 tails in a row on a coin? How much less likely to get this result compared to throwing a single tail?

The chance of throwing a tail followed by a tail, followed by another tail must be much smaller than just throwing one tail. Hence multiplying probabilities.

$$\begin{array}{lll}\hfill P(\text{tail})\text{and}P(\text{tail})\text{and}P(\text{tail})& =\hfill & \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\hfill \\ \hfill & =\hfill & \frac{1}{8}\hfill \\ \hfill & =\hfill & 0.125\hfill \end{array}$$
So you are 4 times less likely to get three successive tails than throwing a single tail ($\frac{0.5}{0.125}=4$).

2. What is the probability of rolling two 5s in succession, on a 6 die?

$$\begin{array}{lll}\hfill P(\text{5})\text{and}P(\text{5})& =\hfill & \frac{1}{6}\times \frac{1}{6}\hfill \\ \hfill & =\hfill & \frac{1}{36}\hfill \\ \hfill & =\hfill & 0.02\dot{7}\hfill \end{array}$$

## ‘AND’ Probability

The probability rule for calculating the probability of outcome of event 1 **AND** event 2 is as follows:
$$P(\text{AandB})=P(A)\times P(B)$$
Event 1 and event 2 **must be** independent of one another. i.e. The outcome of event 1 **does not** effect the probability of event 2.