10.1 Key concepts  (Page 4/4)

Complementary events

The probability of complementary events refers to the probability associated with events not occurring. For example, if $P\left(A\right)=0.25$ , then the probability of $A$ not occurring is the probability associated with all other events in $S$ occurring less the probability of $A$ occurring. This means that

where A' refers to `not A' In other words, the probability of `not A' is equal to one minus the probability of A.

Interpretation of probability values

The probability of an event is generally represented as a real number between 0 and 1, inclusive. An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".

Most probabilities that occur in practice are numbers between 0 and 1, indicating the event's position on the continuum between impossibility 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 or tails-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 all other 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 often written "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 odds are equivalent to probability 3/5.

Summary

• Random experiments
• Outcome
• Sample space
• Event
• Probability models
• Classical theory Relative frequency vs. probability
• Probability identities
• Mutually exclusive events
• Complementary events

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 probability that 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 a Fanta, 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 Opals. What is the probability that the first car to leave the car park is:
   1. an Opal
   2. not an Opal
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 taken out 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 at random, 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 and red 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 the probability 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, the card 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.