Transcript Probability

Probability
The chance that something can
happen
Probability scale



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A probability of 0 means that an event is
impossible
A probability of 1 means that an event is certain
Probability of an event happening is given as a
fraction or decimal
Probability of an event happening is written as
P(x) where x is the event
P(x) = Number of possible outcomes in the event
total number of possible outcomes
Probability Examples

A fair dice is rolled. What is the Probability of
getting:
a) a 6
b) an odd number
c) a 2 or a 3?
There are 6 possible outcomes.
a) P(6)=1/6
there is only one 6
b) P(odd no.)=3/6
1,3,5 are odd numbers
c) P(2 or 3)=2/6
2 and 3 only appear once
Examples

This table shows how
100 counters are
coloured red or blue
and numbered 1 or 2.
The 100 counters are
put in a bag and a
counter is taken from
the bag at random.
Red
Blue
1
23
19
2
32
26
(a)
(b)
Calculate the probability
that the counter is red?
Calculate the probability
that the counter is blue
and numbered 1?
Counters continued
Total number of counters = 23+32+19+26
= 100
Total number of red counters = 23+32
= 55
(a) P(Red)= 55/100
(b) P(Blue and 1)= 19/100
Estimating Probabilities using
Relative Frequency
 The
Relative Frequency of an event is
given by:
Relative Frequency=
No. of times an event happens in an experiment(or survey)
Total No. of trials in the experiment
(or observations in survey)
Relative Frequency Examples
 In
an experiment a pin is dropped for a
100 trials. The drawing pin lands “point
up” 37 times. What is the relative
frequency of the drawing pin landing “point
up”?
Relative Frequency = 37/100
= 0.37
Relative Frequency Examples

A counter was taken from a bag of counters and
replaced. The relative frequency of getting a
blue counter was found to be 0.4. There are 20
counters in the bag. Estimate the number of
blue counters.
Relative frequency of getting a blue counter is 0.4.
Total number of counters in bag = 20
Number of blue counters = 20 x 0.4 = 8.
There are 8 blue counters in the bag.
Mutually Exclusive Events
 These
are events that cannot happen at
the same time.
 For example when you throw a coin you
cannot get a “Heads” at the same time as
a “tails.”
 When A and B are events which cannot
happen at the same time:
P(A or B) = P(A) + P(B)
The probability of an event not
happening
 The
events A and not A cannot happen at
the same time. Because the events A and
not A are certain to happen:
P(not A) = 1 - A
Example

(a)
(b)
(c)
A bag contains 3 red (R) counters, 2 blue (B) counters
and 5 green (G) counters. A counter is taken from the
bag at random. What is the probability that the counter
is:
Red
Green
Red or green?
Total number of counters = 3+2+5=10
(a) P(R) = 3/10
3 red counters
(b) P(G) = 5/10=1/2
5 green counters
(c) P(R or G) = 8/10
3 red +5 green
Example

A bag contains 10 counters. 3 of the counters
are red(R). A counter from the bag is taken at
random. What is the probability that the
counter is:
(a) Red
(b) Not red?
(a)
(b)
P(R) = 3/10
3 red counters out of 10
P(not R) = 1- P(R)
= 1- 3/10
= 7/10
Combining 2 events
 A fair
coin is thrown twice. Write down all
the possible outcomes and their
probabilities.
 A fair
dice is thrown twice. Write down all
the possible outcomes and find the
probability of throwing a double.
Independent Events
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Independent events are events that do not
influence eachother. When two coins are thrown
one result does not effect the other or the person
sitting next to you can swim but this does not
effect your ability to swim or not swim.
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When A and B are independent events then the
probability of A and B occurring is given by
P(A and B) = P(A) x P(B)
P(A and B and C)= P(A) x P(B) x P(C)
Tree diagrams
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(a)
(b)
Box A contains 1 red ball (R) and 1 blue
ball (B). Box B contains 3 red balls (R)
and 2 blue balls (B). A ball is taken at
random from Box A. A ball is then taken
at random from Box B.
Draw a tree diagram to show all possible
outcomes.
Calculate the probability that 2 red balls
are taken.
Tree Diagrams
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(a)
(b)
The probability that Amanda is late for
school is 0.4. Use a tree diagram to find
the probability that on two days running
She is late twice
She is late exactly once