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10.01 Compound events

Lesson

Probability notation

The concepts in probability can often be easier to represent using certain notation.

Instead of writing a word or phrase to talk about the probability of an event like 'flipping a head' or $P\left(\text{head}\right)$P(head), we often represent an event with a single letter, for example, representing the event of flipping a head as $H$H. This can make it is easier to talk about this event. We can then talk about the probability of that event as being $P\left(H\right)$P(H), rather than writing$P\left(\text{head}\right)$P(head) every time.

To investigate some further notation let's consider two events: $A$A and $B$B. $A$A is the event of you seeing an asteroid tonight. $B$B is the event that you will have a burger tonight. A Venn diagram is shown below of the two events with the area of $P(A)$P(A) highlighted.

Venn diagram of event $P(A)$P(A)

 

The complement of the probability of $A$A is given by $1-P\left(A\right)$1P(A). We also have annotation for the complement of the event itself, $A'$A, it may also be written as $\overline{A}?$A?. Given our event ,$A$A, the event $A'$A would be the event not seeing an asteroid tonight.

Venn diagram of event $P(A')$P(A)

 

The notation $\cup$ used as $P\left(A\cup B\right)$P(AB) means the probabilty of $A$A or $B$B happen.

It is the probability of

  • you seeing an asteroid but not eating a burger
  • you eating a burger but not seeing an asteroid
  • you seeing an asteroid and eating a burger

Venn diagram of event $P(A\cup B)$P(AB)

 

The notation $\cap$ used as $P\left(A\cap B\right)$P(AB) means the probabilty of $A$A and $B$B both happen.

It is the probability of

  • you seeing an asteroid and eating a burger

Venn diagram of event $P(A\cap B)$P(AB)

 

Venn diagrams are a useful way to think about these new notations, below are some highlighted Venn diagrams with the notation for the area written below:

Summary

 

  • $A$A - a single letter can represent an event.
  • $A'$A - the complement of event $A$A.
  • $P\left(A\right)$P(A) - the probability of event $A$A happening.
  • $P\left(A\cup B\right)$P(AB) - the probability that either of $A$A or $B$B happening.
  • $P\left(A\cap B\right)$P(AB) - the probability that both $A$A and $B$B happening.

 

Practice questions

Question 1

Two events are defined as:

Event $A$A: it will rain tomorrow

Event $B$B: there will be a storm tomorrow

The notation $P(B)$P(B) is suitable to describe which of the following probabilities?

  1. probability of there being a storm tomorrow but no rain

    A

    probability of no rain tomorrow

    B

    probability of there being either a storm or rain tomorrow

    C

    probability of a storm occurring tomorrow

    D

    probability of there being a storm tomorrow but no rain

    A

    probability of no rain tomorrow

    B

    probability of there being either a storm or rain tomorrow

    C

    probability of a storm occurring tomorrow

    D
Question 2

In an experiment, a number is chosen randomly from the numbers listed below:

{$2$2, $3$3, $5$5, $6$6, $7$7, $10$10, $12$12, $14$14, $15$15, $16$16, $19$19, $20$20}

Event $A$A = odd number is chosen

Event $B$B = multiple of $4$4 is chosen

  1. Which of the following has the largest probability?

    $B'$B

    A

    $A\cup B$AB

    B

    $B$B

    C

    $A\cap B$AB

    D

    $A$A

    E

    $B'$B

    A

    $A\cup B$AB

    B

    $B$B

    C

    $A\cap B$AB

    D

    $A$A

    E
  2. Which of the following has a value of $0$0?

    $P\left(A\cap B\right)$P(AB)

    A

    $P\left(A'\cup B\right)$P(AB)

    B

    $P((A\cap B)')$P((AB))

    C

    $P\left(A'\cap B'\right)$P(AB)

    D

    $P\left(A\cap B\right)$P(AB)

    A

    $P\left(A'\cup B\right)$P(AB)

    B

    $P((A\cap B)')$P((AB))

    C

    $P\left(A'\cap B'\right)$P(AB)

    D
Question 3

A student creates the following diagram of their favourite animals.

The event $F$F is: "selecting a favourite four legged animal".

The event $S$S is : "selecting a favourite animal with stripes".

  1. Which region represents the favourable outcomes for the probability $P\left(F\right)$P(F)?

    A

    B

    C

    D

    A

    B

    C

    D
  2. Which region represents the favourable outcomes for the probability $P(S')$P(S)?

    A

    B

    C

    D

    A

    B

    C

    D

 

Outcomes

VCMSP347

Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence.

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