10.1 Probability: part 2  (Page 2/2)

For example, we can say that the sun will always rise in the east. This is a certain event, the sun will not suddenly rise in the north. But if we looked at the event of Penny Heyns winning a swimming race against your maths teacher, then this event is almost certain, since there is a very small chance that your teacher could win the race.

Most probabilities that occur in practice are numbers between 0 and 1, indicating the event's position on the continuum betweenimpossibility and certainty. The closer an event's probability is to 1, the more likely it is to occur.

For example, if two mutually exclusive events are assumed equally probable, such as a flipped or spun coin landing heads-up ortails-up, we can express the probability of each event as "1 in 2", or, equivalently, "50%" or "1/2".

Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of allother events. The odds of heads-up, for the tossed/spun coin, are (1/2)/(1 - 1/2), which is equal to 1/1. This is expressed as "1 to 1 odds" and oftenwritten "1:1".

Odds a:b for some event are equivalent to probability a/(a+b). For example, 1:1 odds are equivalent to probability 1/2, and 3:2 oddsare equivalent to probability 3/5.

Summary

• The term random experiment or statistical experiment is used to describe any repeatable process, the results of which are analyzed in some way.
• An outcome of an experiment is a single result of that experiment.
• The sample space of an experiment is the complete set of possible outcomes of the experiment.
• An event is any set of outcomes of an experiment.
• A Venn diagram can be used to show the relationship between the possible outcomes of a random experiment and the sample space. Venn diagrams can also beused to indicate the union and intersection between events in a sample space.
• When all outcomes are equally likely, they have an equal chance of happening. $P\left(E\right)=n\left(E\right)/n\left(S\right)$ gives us the probability of an equally likely outcome happening.
• Relative frequency is defined as the number of times an event happens in a statistical experiment divided by thenumber of trials conducted.
• The following results apply to probabilities, for the sample space $S$ and two events $A$ and $B$ , within $S$ .
  $$P\left(S\right)=1$$
  $$P(A\cap B)=P\left(A\right)\times P\left(B\right)$$
  $$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$$
• Mutually exclusive events are events, which cannot be true at the same time.
• $$P\left(A^{\text{'}}\right)=1-P\left(A\right)$$ is the probability that A will not occur. This is known as a complementary event.

We can summarize some of the key concepts in this chapter in the following table:

Term	Meaning	Representation	Venn diagram
Union	Everything in A and B	$A\cup B$
Intersection	Everything in A or B	$A\cap B$
Complement	Everything that is not in A	$A_c$
Only one	All that is only in A	$A-B$

End of chapter exercises

1. A group of 45 children were asked if they eat Frosties and/or Strawberry Pops. 31 eat both and 6 eat only Frosties. What is the probabilitythat a child chosen at random will eat only Strawberry Pops?
   
2. In a group of 42 pupils, all but 3 had a packet of chips or a Fanta or both. If 23 had a packet of chips and 7 of these also had aFanta, what is the probability that one pupil chosen at random has:
   1. Both chips and Fanta
   2. has only Fanta?
3. Use a Venn diagram to work out the following probabilities from a die being rolled:
   1. A multiple of 5 and an odd number
   2. a number that is neither a multiple of 5 nor an odd number
   3. a number which is not a multiple of 5, but is odd.
4. A packet has yellow and pink sweets. The probability of taking out a pink sweet is 7/12.
   1. What is the probability of taking out a yellow sweet
   2. If 44 if the sweets are yellow, how many sweets are pink?
5. In a car park with 300 cars, there are 190 Opels. What is the probability that the first car to leave the car park is:
   1. an Opel
   2. not an Opel
6. Tamara has 18 loose socks in a drawer. Eight of these are orange and two are pink. Calculate the probability that the first sock takenout at random is:
   1. Orange
   2. not orange
   3. pink
   4. not pink
   5. orange or pink
   6. not orange or pink
7. A plate contains 9 shortbread cookies, 4 ginger biscuits, 11 chocolate chip cookies and 18 Jambos. If a biscuit is selected atrandom, what is the probability that:
   1. it is either a ginger biscuit of a Jambo?
   2. it is NOT a shortbread cookie.
8. 280 tickets were sold at a raffle. Ingrid bought 15 tickets. What is the probability that Ingrid:
   1. Wins the prize
   2. Does not win the prize?
9. The children in a nursery school were classified by hair and eye colour. 44 had red hair and not brown eyes, 14 had brown eyes andred hair, 5 had brown eyes but not red hair and 40 did not have brown eyes or red hair.
   1. How many children were in the school
   2. What is the probility that a child chosen at random has:
      1. Brown eyes
      2. Red hair
   3. A child with brown eyes is chosen randomly. What is the probability that this child will have red hair
10. A jar has purple, blue and black sweets in it. The probability that a sweet, chosen at random, will be purple is 1/7 and theprobability that it will be black is 3/5.
    1. If I choose a sweet at random what is the probability that it will be:
       1. purple or blue
       2. Black
       3. purple
    2. If there are 70 sweets in the jar how many purple ones are there?
    3. 1/4 if the purple sweets in b) have streaks on them and rest do not. How many purple sweets have streaks?
11. For each of the following, draw a Venn diagram to represent the situation and find an example to illustrate the situation.
    1. A sample space in which there are two events that are not mutually exclusive
    2. A sample space in which there are two events that are complementary.
12. Use a Venn diagram to prove that the probability of either event A or B occuring is given by: (A and B are not exclusive)P(A or B) = P(A) + P(B) - P(A and B)
    
13. All the clubs are taken out of a pack of cards. The remaining cards are then shuffled and one card chosen. After being chosen, thecard is replaced before the next card is chosen.
    1. What is the sample space?
    2. Find a set to represent the event, P, of drawing a picture card.
    3. Find a set for the event, N, of drawing a numbered card.
    4. Represent the above events in a Venn diagram
    5. What description of the sets P and N is suitable? (Hint: Find any elements of P in N and N in P.)
14. Thuli has a bag containing five orange, three purple and seven pink blocks. The bag is shaken and a block is withdrawn. The colour of the block is noted and the block is replaced.
    1. What is the sample space for this experiment?
    2. What is the set describing the event of drawing a pink block, P?
    3. Write down a set, O or B, to represent the event of drawing either a orange or a purple block.
    4. Draw a Venn diagram to show the above information.