14E

Multi-stage experiments and the product rule

Independent events, the product rule P(A ∩ B) = P(A) × P(B) for multi-stage experiments, and "at least one" problems.

The product rule

Events are independent if the outcome of one does not change the probability of the other, such as two separate dice rolls or coin tosses.

Product rule for independent events

$$P(A \cap B) = P(A) \times P(B)$$

This extends to any number of stages: multiply the probability of each stage.

Example 1●

A coin is tossed and a die is rolled. Find $P(\text{head and } 6)$.

$\dfrac12 \times \dfrac16 = \dfrac{1}{12}$

Example 2: at least one●●

A die is rolled four times. Find the probability of at least one six, to 3 decimal places.

$P(\text{no six on one roll}) = \dfrac56$, so $P(\text{no sixes}) = \left(\dfrac56\right)^4 = \dfrac{625}{1296}$.

$P(\text{at least one six}) = 1 - \dfrac{625}{1296} = \dfrac{671}{1296} \approx 0.518$

Flashcards

Independent events

One does not affect the other's probability.

Product rule

$P(A \cap B) = P(A)P(B)$

$P( ext{at least one})$

$1 - P(\text{none})$

Practice

Find $P(\text{head and } 6)$ when a coin is tossed and a die rolled.
Three dice are rolled. Find $P(\text{three sixes})$.
A die is rolled twice. Find $P(\text{no sixes})$.
Find $P(\text{at least one six})$ in two rolls.
Events $A$ and $B$ are independent when:
The chance of rain is $0.3$ each day, independently. Find $P(\text{rain on both of two days})$.
Find $P(\text{no rain on either day})$.
Find the probability of at least one six in four rolls of a die, to 3 decimal places.
An archer hits the target with probability $0.8$ each shot, independently. Find $P(\text{hits with all three shots})$.