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Probability and Venn diagrams
SLIDE 1
? Write out explicitly the sets S and P.
? Let S be the domain of discourse. Draw a Venn diagram showing the sets S and P.
? is the order of the set S. Find and .
? Find p, where p is the probability of choosing a prime number from S.
? Find . What is the relation p and q?
SLIDE 2
?
?
? ,
? Probability prime,
?
, they are both a statement of what probability means in this context.
SLIDE 3
? Write out explicitly the sets V and S.
? Let V be the domain of discourse. Draw a Venn diagram showing the sets B and S.
? Find and .
? Find . What is this number?
SLIDE 4
?
?
? ,
?
SLIDE 5
Events
Let V be a set of events, and let A be a subset of these events.
A is an event of type A belonging to the set of events of type V.
Let be the number of events V. Let be the number of events of type A.
Probability
Then the probability of A is , where .
SLIDE 6
A die is rolled once and the number facing upwards is recorded. This is called a trial. The result is an event called an outcome.
? Let V be the set of all possible outcomes from the trial. List all the possible outcomes that are events in V.
? Let A be the set of all possible outcomes that are even numbers. List all the possible outcomes in A.
? Use the definition, , to find the probability of an event of type A.
SLIDE 7
A die is rolled once and the number facing upwards is recorded. This is called a trial. The result is an event called an outcome.
? V is the set of all possible outcomes from the trial.
where 1 is short for “the number facing upward on the die is a 1” and so forth.
? A is the set of all possible outcomes that are even numbers.
? probability of A,
SLIDE 8
There are 52 cards in a pack of playing cards. There are four suits in a pack.
clubs diamonds hearts spades
A trial consists of taking one card from a pack of 52 playing cards.
? Let V be the set of all possible outcomes from the trial. What is ?
? Let A be the event that a heart is chosen. List all the possible outcomes in A. Find .
? Find , the probability of A.
SLIDE 9
clubs diamonds hearts spades
A trial consists of taking one card from a pack of 52 playing cards.
? There are 52 cards in a pack .
? Let A be the event that a heart is chosen.
where 2 is short for “2 of hearts”.
?
SLIDE 10
Intersection of two sets
is the set of all numbers that are members of both A and B. is the intersection of sets A and B.
? Write out explicitly the sets V, A, B and
? Let V be the domain of discourse. Draw a Venn diagram showing the sets V, A, B and .
? Find , , and .
? Find the probability of .
SLIDE 11
?
? , , ,
?
SLIDE 12
Union of two sets
is the set of all numbers that are members of either A or B. is the union of sets A and B.
? Let V be the domain of discourse. Draw a Venn diagram showing the sets V, A, B and .
? Find , , and .
? Find the probability of .
SLIDE 13
?
? , , ,
?
SLIDE 14
Mutually exclusive
Two events are mutually exclusive if you cannot have both together from a single trial.
In a Venn diagram, when two sets of possible outcomes (two “events”) are mutually exclusive, their bubbles have no overlap.
? Let V be the domain of discourse. Draw a Venn diagram showing the sets V, A and B.
? Find , , and .
? Find the probability of .
SLIDE 15
?
? , , ,
?
SLIDE 16
is the empty or null set.
Suppose that A and B are mutually exclusive outcomes of two types belonging to a domain of possible events. Consider the following three statements.
I
II
III
Which of these statements are true?
A I only
B I and II only
C I and III only
D I, II and III
E none
SLIDE 17
is the empty or null set.
If A and B are mutually exclusive outcomes of two types belonging to a domain of possible events.
Then the following statements are all true.
I
II
III
D I, II and III
SLIDE 18
60 English-speaking adults were asked if they speak Spanish or French.
20 said they speak French. Of these, 4 said they speak both Spanish and French. 30 said they do not speak either Spanish or French.
Denote the event that the adult speaks Spanish by S, and the event that the adult speaks French by F.
? Make a Venn diagram of the information, showing on the Venn diagram , and the number that speaking neither French nor Spanish.
? Find and . Add these numbers to the Venn diagram.
? Find out the probability that two of the adults chosen at random speak only Spanish.
SLIDE 19
?
, ,
?
? To find the probability that two of the adults chosen at random speak only Spanish. This is a probability problem without replacement, since the two adults must be different adults. Of the 60 adults, there are 10 who speak only Spanish, so the probability of choosing the first is . The probability of choosing the second is . Let the two adults chosen be A and B, then we could have chosen A first then B or B first then A, so the probability of choosing two adults at random who speak only Spanish is .
SLIDE 20
A box contains four kinds of counter: black, red, black and red, and neither black nor red (white).
There are 50 counters in the box. Of these 12 are wholly or partly coloured black, 18 contain the colour red, and 26 are neither black nor red (they are white).
Denote the event that the counter is all or partly coloured black by B, and the event that the counter is all or partly coloured red by R.
? Make a Venn diagram of the information, showing on the Venn diagram , and the number of counters that are neither black nor red.
? Find and
? A counter is drawn from the box and whether it contains black, red or both colours is noted. The counter is put back into the box, and a second counter is drawn. Find the probability that both counters taken from the black will contain only the colour red.
SLIDE 21
?
?
Let , then
? Since the counter is put back into the box, this is probability with replacement. The probability of taking a counter that is only red is . The probability of choosing the second is also . Let the two counters chosen be A and B, then we could have chosen A first then B or B first then A, so the probability of choosing two counters that are wholly red is .
SLIDE 22
50 people were asked to select the kind of movie they watch from a list of Action, Comedy or Melodrama.
Of these people
29 said the watch Action
2 said they watch Action, Comedy and Melodrama
5 said they watch Action and Melodrama but not Comedy
7 said they watch Comedy and Melodrama
9 said they do not watch any of these kind of movie
all 12 people who said they watch Comedy, said they watch at least one other of the three kinds of movie.
Two of the 50 people are chosen at random. Work out the probability that they both only watch Melodrama.
SLIDE 23
The information in the question translates into the above Venn diagram, where the numbers in green are totals for a bubble. We can then fill in all the remaining numbers.
The number watching only Melodrama is 7.
The probability problem is without replacement, so the probability is
SLIDE 24
50 people were asked whether they had visited any of the following destinations on holiday: New York, Paris, Berlin.
Of these people
16 said they had visited New York
3 said they had visited all three
4 said they had visited New York and Berlin only
5 said they had visited Paris and Berlin but not New York
Of the 18 that said they had visited Paris, 8 had not visited either New York or Berlin
12 said they had not visited any of the three cities.
Two of the 50 people are chosen at random. Work out the probability that one had visited only New York and the other had visited only Berlin.
SLIDE 25
The information in the question translates into the above Venn diagram, where the numbers in green are totals for a bubble. We can then fill in all the remaining numbers.
The number that had visited only New York was 7, and the number that had visited only Berlin was 9.
The probability problem is without replacement. Denote the probability that one had visited only New York and the other had visited only Berlin by . Then
SLIDE 26
The Venn diagram above shows the number of adults over the age of 40 who have at some time in their life voted for one of the parties X, Y or Z.
? Find the probability that the adult has voted for more than one party
? Political opinions in this country are such that no one who has voted to party X has ever voted for party Z. What kind of events are voting for X and voting for Z?
Two adults are chosen at random.
? Find the probability that both have voted for at most one parties.
SLIDE 27
? There are 36 adults in the sample, and have voted for more than one party. The probability that the adult has voted for more than one party is .
? Voting for X and voting for Z are mutually exclusive events.
? The number of people who have voted for at most only one party is . The problem of finding the probability that two people have voted for at most one party is a problem without replacement, as the two people must be different. This probability is
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