10.1 Key concepts  (Page 3/4)

Project idea

Perform an experiment to show that as the number of trials increases, the relative frequency approaches the probability of a coin toss. Perform 10, 20, 50, 100, 200 trials of tossing a coin.

Probability identities

The following results apply to probabilities, for the sample space $S$ and two events $A$ and $B$ , within $S$ .

The following video provides a brief summary of some of the work covered so far.

Probability identities

Answer the following questions

1. Rory is target shooting. His probability of hitting the target is $\mathrm{0,7}$ . He fires five shots. What is the probability that all five shots miss the center?
   
2. An archer is shooting arrows at a bullseye. The probability that an arrow hits the bullseye is $\mathrm{0,4}$ . If she fires three arrows, what is the probability that all the arrows hit the bullseye?
   
3. A dice with the numbers 1,3,5,7,9,11 on it is rolled. Also a fair coin is tossed. What is the probability that:
   1. A tail is tossed and a 9 rolled?
   2. A head is tossed and a 3 rolled?
4. Four children take a test. The probability of each one passing is as follows. Sarah: $\mathrm{0,8}$ , Kosma: $\mathrm{0,5}$ , Heather: $\mathrm{0,6}$ , Wendy: $\mathrm{0,9}$ . What is the probability that:
   1. all four pass?
   2. all four fail?
5. With a single pick from a pack of 52 cards what is the probability that the card will be an ace or a black card?
   

Mutually exclusive events

Mutually exclusive events are events, which cannot be true at the same time.

Examples of mutually exclusive events are:

1. A die landing on an even number or landing on an odd number.
2. A student passing or failing an exam
3. A tossed coin landing on heads or landing on tails

This means that if we examine the elements of the sets that make up $A$ and $B$ there will be no elements in common. Therefore, $A\cap B=\varnothing$ (where $\varnothing$ refers to the empty set). Since, $P(A\cap B)=0$ , equation [link] becomes:

for mutually exclusive events.

Mutually exclusive events

Answer the following questions

1. A box contains coloured blocks. The number of each colour is given in the following table.

   Colour	Purple	Orange	White	Pink
   Number of blocks	24	32	41	19

   A block is selected randomly. What is the probability that the block will be:
   1. purple
   2. purple or white
   3. pink and orange
   4. not orange?
2. A small private school has a class with children of various ages. The table gies the number of pupils of each age in the class.

   3 years female	3 years male	4 years female	4 years male	5 years female	5 years male
   6	2	5	7	4	6

   If a pupil is selceted at random what is the probability that the pupil will be:
   1. a female
   2. a 4 year old male
   3. aged 3 or 4
   4. aged 3 and 4
   5. not 5
   6. either 3 or female?
3. Fiona has 85 labeled discs, which are numbered from 1 to 85. If a disc is selected at random what is the probability that the disc number:
   1. ends with 5
   2. can be multiplied by 3
   3. can be multiplied by 6
   4. is number 65
   5. is not a multiple of 5
   6. is a multiple of 4 or 3
   7. is a multiple of 2 and 6
   8. is number 1?